Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 17.4s

Alternatives: 17

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

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

FPCore

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

Julia

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

Wolfram

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

Derivation

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. +-commutative N/A

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

   2. +-commutative N/A

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

   3. associate--l+ N/A

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

   4. associate-+r+ N/A

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

   5. +-lowering-+.f64 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. +-lowering-+.f64 N/A

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

   9. metadata-eval N/A

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

   10. log-lowering-log.f64 N/A

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

   11. log-lowering-log.f64 N/A

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

   12. --lowering--.f64 N/A

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

   13. log-lowering-log.f64 N/A

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

   14. +-lowering-+.f64 99.5

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

4. Applied egg-rr 99.5%

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

Alternative 2: 88.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right)\\ t_2 := \left(\left(\log z + t\_1\right) - t\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t\_2 \leq -20000:\\ \;\;\;\;\left(t\_1 - t\right) + a \cdot \log t\\ \mathbf{elif}\;t\_2 \leq 870:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(z \cdot \left(x + y\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ x y)))
        (t_2 (+ (- (+ (log z) t_1) t) (* (log t) (- a 0.5)))))
   (if (<= t_2 -20000.0)
     (+ (- t_1 t) (* a (log t)))
     (if (<= t_2 870.0)
       (- (fma -0.5 (log t) (log (* z (+ x y)))) t)
       (+ (log y) (fma (log t) (+ a -0.5) (- t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y));
	double t_2 = ((log(z) + t_1) - t) + (log(t) * (a - 0.5));
	double tmp;
	if (t_2 <= -20000.0) {
		tmp = (t_1 - t) + (a * log(t));
	} else if (t_2 <= 870.0) {
		tmp = fma(-0.5, log(t), log((z * (x + y)))) - t;
	} else {
		tmp = log(y) + fma(log(t), (a + -0.5), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	t_2 = Float64(Float64(Float64(log(z) + t_1) - t) + Float64(log(t) * Float64(a - 0.5)))
	tmp = 0.0
	if (t_2 <= -20000.0)
		tmp = Float64(Float64(t_1 - t) + Float64(a * log(t)));
	elseif (t_2 <= 870.0)
		tmp = Float64(fma(-0.5, log(t), log(Float64(z * Float64(x + y)))) - t);
	else
		tmp = Float64(log(y) + fma(log(t), Float64(a + -0.5), Float64(-t)));
	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[(N[(N[(N[Log[z], $MachinePrecision] + t$95$1), $MachinePrecision] - t), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -20000.0], N[(N[(t$95$1 - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 870.0], N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + (-t)), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -2e4

   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. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. associate-+r+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. +-lowering-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. log-lowering-log.f64 N/A

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

      11. log-lowering-log.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. log-lowering-log.f64 N/A

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

      14. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 98.6

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

   7. Simplified 98.6%

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

   if -2e4 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 870

   1. Initial program 98.6%

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

      1. +-commutative N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sub-neg N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, \mathsf{neg}\left(\color{blue}{\frac{1}{4}}\right)\right) \cdot \log t, \frac{1}{a + \frac{1}{2}}, \left(\log \left(x + y\right) + \log z\right) - t\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, \color{blue}{\frac{-1}{4}}\right) \cdot \log t, \frac{1}{a + \frac{1}{2}}, \left(\log \left(x + y\right) + \log z\right) - t\right) \]

      11. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, \frac{-1}{4}\right) \cdot \color{blue}{\log t}, \frac{1}{a + \frac{1}{2}}, \left(\log \left(x + y\right) + \log z\right) - t\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, \frac{-1}{4}\right) \cdot \log t, \color{blue}{\frac{1}{a + \frac{1}{2}}}, \left(\log \left(x + y\right) + \log z\right) - t\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, \frac{-1}{4}\right) \cdot \log t, \frac{1}{\color{blue}{a + \frac{1}{2}}}, \left(\log \left(x + y\right) + \log z\right) - t\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, \frac{-1}{4}\right) \cdot \log t, \frac{1}{a + \frac{1}{2}}, \color{blue}{\left(\log \left(x + y\right) + \log z\right) - t}\right) \]

      15. sum-log N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, \frac{-1}{4}\right) \cdot \log t, \frac{1}{a + \frac{1}{2}}, \color{blue}{\log \left(\left(x + y\right) \cdot z\right)} - t\right) \]

      16. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, \frac{-1}{4}\right) \cdot \log t, \frac{1}{a + \frac{1}{2}}, \color{blue}{\log \left(\left(x + y\right) \cdot z\right)} - t\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, \frac{-1}{4}\right) \cdot \log t, \frac{1}{a + \frac{1}{2}}, \log \color{blue}{\left(\left(x + y\right) \cdot z\right)} - t\right) \]

      18. +-lowering-+.f64 92.2

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, a, -0.25\right) \cdot \log t, \frac{1}{a + 0.5}, \log \left(\color{blue}{\left(x + y\right)} \cdot z\right) - t\right) \]

   4. Applied egg-rr 92.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, a, -0.25\right) \cdot \log t, \frac{1}{a + 0.5}, \log \left(\left(x + y\right) \cdot z\right) - t\right)} \]

   5. Taylor expanded in a around 0

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

      1. --lowering--.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. log-lowering-log.f64 N/A

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

      5. log-lowering-log.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-lowering-+.f64 88.4

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

   7. Simplified 88.4%

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

   if 870 < (+.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 x around 0

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. log-lowering-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-lowering-+.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. log-lowering-log.f64 69.5

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

   5. Simplified 69.5%

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

   6. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 62.1

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

   8. Simplified 62.1%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log z + \log \left(x + y\right)\right) - t\right) + \log t \cdot \left(a - 0.5\right) \leq -20000:\\ \;\;\;\;\left(\log \left(x + y\right) - t\right) + a \cdot \log t\\ \mathbf{elif}\;\left(\left(\log z + \log \left(x + y\right)\right) - t\right) + \log t \cdot \left(a - 0.5\right) \leq 870:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(z \cdot \left(x + y\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right)\\ t_2 := \left(\left(\log z + t\_1\right) - t\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t\_2 \leq -20000:\\ \;\;\;\;\left(t\_1 - t\right) + a \cdot \log t\\ \mathbf{elif}\;t\_2 \leq 870:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ x y)))
        (t_2 (+ (- (+ (log z) t_1) t) (* (log t) (- a 0.5)))))
   (if (<= t_2 -20000.0)
     (+ (- t_1 t) (* a (log t)))
     (if (<= t_2 870.0)
       (- (fma (log t) -0.5 (log (* z y))) t)
       (+ (log y) (fma (log t) (+ a -0.5) (- t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y));
	double t_2 = ((log(z) + t_1) - t) + (log(t) * (a - 0.5));
	double tmp;
	if (t_2 <= -20000.0) {
		tmp = (t_1 - t) + (a * log(t));
	} else if (t_2 <= 870.0) {
		tmp = fma(log(t), -0.5, log((z * y))) - t;
	} else {
		tmp = log(y) + fma(log(t), (a + -0.5), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	t_2 = Float64(Float64(Float64(log(z) + t_1) - t) + Float64(log(t) * Float64(a - 0.5)))
	tmp = 0.0
	if (t_2 <= -20000.0)
		tmp = Float64(Float64(t_1 - t) + Float64(a * log(t)));
	elseif (t_2 <= 870.0)
		tmp = Float64(fma(log(t), -0.5, log(Float64(z * y))) - t);
	else
		tmp = Float64(log(y) + fma(log(t), Float64(a + -0.5), Float64(-t)));
	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[(N[(N[(N[Log[z], $MachinePrecision] + t$95$1), $MachinePrecision] - t), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -20000.0], N[(N[(t$95$1 - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 870.0], N[(N[(N[Log[t], $MachinePrecision] * -0.5 + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + (-t)), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -2e4

   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. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. associate-+r+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. +-lowering-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. log-lowering-log.f64 N/A

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

      11. log-lowering-log.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. log-lowering-log.f64 N/A

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

      14. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 98.6

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

   7. Simplified 98.6%

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

   if -2e4 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 870

   1. Initial program 98.6%

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

      1. flip-+ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. difference-of-squares N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sum-log N/A

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

      6. log-lowering-log.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. diff-log N/A

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

      10. log-lowering-log.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. +-lowering-+.f64 N/A

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

      13. diff-log N/A

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

      14. log-lowering-log.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. +-lowering-+.f64 60.2

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

   4. Applied egg-rr 60.2%

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

   5. Taylor expanded in x around 0

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

      1. --lowering--.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. log-lowering-log.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 45.2

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

   7. Simplified 45.2%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 44.2%

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

   if 870 < (+.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 x around 0

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. log-lowering-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-lowering-+.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. log-lowering-log.f64 69.5

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

   5. Simplified 69.5%

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

   6. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 62.1

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

   8. Simplified 62.1%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log z + \log \left(x + y\right)\right) - t\right) + \log t \cdot \left(a - 0.5\right) \leq -20000:\\ \;\;\;\;\left(\log \left(x + y\right) - t\right) + a \cdot \log t\\ \mathbf{elif}\;\left(\left(\log z + \log \left(x + y\right)\right) - t\right) + \log t \cdot \left(a - 0.5\right) \leq 870:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log z + \log \left(x + y\right)\right) - t\right) + \log t \cdot \left(a - 0.5\right)\\ t_2 := a \cdot \log t - t\\ \mathbf{if}\;t\_1 \leq -10000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2000:\\ \;\;\;\;\log y - t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log z) (log (+ x y))) t) (* (log t) (- a 0.5))))
        (t_2 (- (* a (log t)) t)))
   (if (<= t_1 -10000.0) t_2 (if (<= t_1 2000.0) (- (log y) t) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log(z) + log((x + y))) - t) + (log(t) * (a - 0.5));
	double t_2 = (a * log(t)) - t;
	double tmp;
	if (t_1 <= -10000.0) {
		tmp = t_2;
	} else if (t_1 <= 2000.0) {
		tmp = log(y) - t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((log(z) + log((x + y))) - t) + (log(t) * (a - 0.5d0))
    t_2 = (a * log(t)) - t
    if (t_1 <= (-10000.0d0)) then
        tmp = t_2
    else if (t_1 <= 2000.0d0) then
        tmp = log(y) - t
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((Math.log(z) + Math.log((x + y))) - t) + (Math.log(t) * (a - 0.5));
	double t_2 = (a * Math.log(t)) - t;
	double tmp;
	if (t_1 <= -10000.0) {
		tmp = t_2;
	} else if (t_1 <= 2000.0) {
		tmp = Math.log(y) - t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((math.log(z) + math.log((x + y))) - t) + (math.log(t) * (a - 0.5))
	t_2 = (a * math.log(t)) - t
	tmp = 0
	if t_1 <= -10000.0:
		tmp = t_2
	elif t_1 <= 2000.0:
		tmp = math.log(y) - t
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(z) + log(Float64(x + y))) - t) + Float64(log(t) * Float64(a - 0.5)))
	t_2 = Float64(Float64(a * log(t)) - t)
	tmp = 0.0
	if (t_1 <= -10000.0)
		tmp = t_2;
	elseif (t_1 <= 2000.0)
		tmp = Float64(log(y) - t);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((log(z) + log((x + y))) - t) + (log(t) * (a - 0.5));
	t_2 = (a * log(t)) - t;
	tmp = 0.0;
	if (t_1 <= -10000.0)
		tmp = t_2;
	elseif (t_1 <= 2000.0)
		tmp = log(y) - t;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[z], $MachinePrecision] + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -10000.0], t$95$2, If[LessEqual[t$95$1, 2000.0], N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -1e4 or 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 99.8%

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

      1. flip-+ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. difference-of-squares N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sum-log N/A

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

      6. log-lowering-log.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. diff-log N/A

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

      10. log-lowering-log.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. +-lowering-+.f64 N/A

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

      13. diff-log N/A

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

      14. log-lowering-log.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. +-lowering-+.f64 50.2

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

   4. Applied egg-rr 50.2%

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

   5. Taylor expanded in x around 0

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

      1. --lowering--.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. log-lowering-log.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 63.3

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

   7. Simplified 63.3%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 98.5

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

   10. Simplified 98.5%

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

   if -1e4 < (+.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 98.8%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. log-lowering-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-lowering-+.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. log-lowering-log.f64 50.4

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

   5. Simplified 50.4%

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

   6. Taylor expanded in t around inf

      \[\leadsto \log y + \color{blue}{-1 \cdot t} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 9.9

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

   8. Simplified 9.9%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log z + \log \left(x + y\right)\right) - t\right) + \log t \cdot \left(a - 0.5\right) \leq -10000:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;\left(\left(\log z + \log \left(x + y\right)\right) - t\right) + \log t \cdot \left(a - 0.5\right) \leq 2000:\\ \;\;\;\;\log y - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right)\\ t_2 := \log z + t\_1\\ \mathbf{if}\;t\_2 \leq -720:\\ \;\;\;\;\left(t\_1 - t\right) + a \cdot \log t\\ \mathbf{elif}\;t\_2 \leq 695:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, \log \left(z \cdot \left(x + y\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ x y))) (t_2 (+ (log z) t_1)))
   (if (<= t_2 -720.0)
     (+ (- t_1 t) (* a (log t)))
     (if (<= t_2 695.0)
       (fma (+ a -0.5) (log t) (- (log (* z (+ x y))) t))
       (+ (log y) (fma (log t) (+ a -0.5) (- t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y));
	double t_2 = log(z) + t_1;
	double tmp;
	if (t_2 <= -720.0) {
		tmp = (t_1 - t) + (a * log(t));
	} else if (t_2 <= 695.0) {
		tmp = fma((a + -0.5), log(t), (log((z * (x + y))) - t));
	} else {
		tmp = log(y) + fma(log(t), (a + -0.5), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	t_2 = Float64(log(z) + t_1)
	tmp = 0.0
	if (t_2 <= -720.0)
		tmp = Float64(Float64(t_1 - t) + Float64(a * log(t)));
	elseif (t_2 <= 695.0)
		tmp = fma(Float64(a + -0.5), log(t), Float64(log(Float64(z * Float64(x + y))) - t));
	else
		tmp = Float64(log(y) + fma(log(t), Float64(a + -0.5), Float64(-t)));
	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[(N[Log[z], $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -720.0], N[(N[(t$95$1 - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 695.0], N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + (-t)), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -720

   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. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. associate-+r+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. +-lowering-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. log-lowering-log.f64 N/A

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

      11. log-lowering-log.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. log-lowering-log.f64 N/A

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

      14. +-lowering-+.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 91.8

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

   7. Simplified 91.8%

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

   if -720 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 695

   1. Initial program 99.4%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. +-lowering-+.f64 N/A

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

      5. metadata-eval N/A

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

      6. log-lowering-log.f64 N/A

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

      7. --lowering--.f64 N/A

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

      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. log-lowering-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. *-lowering-*.f64 N/A

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

      11. +-lowering-+.f64 99.6

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

   4. Applied egg-rr 99.6%

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

   if 695 < (+.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 x around 0

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. log-lowering-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-lowering-+.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. log-lowering-log.f64 74.6

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

   5. Simplified 74.6%

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

   6. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 61.7

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

   8. Simplified 61.7%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(x + y\right) \leq -720:\\ \;\;\;\;\left(\log \left(x + y\right) - t\right) + a \cdot \log t\\ \mathbf{elif}\;\log z + \log \left(x + y\right) \leq 695:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, \log \left(z \cdot \left(x + y\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right)\\ t_2 := \log z + t\_1\\ \mathbf{if}\;t\_2 \leq -720:\\ \;\;\;\;\left(t\_1 - t\right) + a \cdot \log t\\ \mathbf{elif}\;t\_2 \leq 695:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, \log \left(z \cdot \left(x + y\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ x y))) (t_2 (+ (log z) t_1)))
   (if (<= t_2 -720.0)
     (+ (- t_1 t) (* a (log t)))
     (if (<= t_2 695.0)
       (- (fma (+ a -0.5) (log t) (log (* z (+ x y)))) t)
       (+ (log y) (fma (log t) (+ a -0.5) (- t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y));
	double t_2 = log(z) + t_1;
	double tmp;
	if (t_2 <= -720.0) {
		tmp = (t_1 - t) + (a * log(t));
	} else if (t_2 <= 695.0) {
		tmp = fma((a + -0.5), log(t), log((z * (x + y)))) - t;
	} else {
		tmp = log(y) + fma(log(t), (a + -0.5), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	t_2 = Float64(log(z) + t_1)
	tmp = 0.0
	if (t_2 <= -720.0)
		tmp = Float64(Float64(t_1 - t) + Float64(a * log(t)));
	elseif (t_2 <= 695.0)
		tmp = Float64(fma(Float64(a + -0.5), log(t), log(Float64(z * Float64(x + y)))) - t);
	else
		tmp = Float64(log(y) + fma(log(t), Float64(a + -0.5), Float64(-t)));
	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[(N[Log[z], $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -720.0], N[(N[(t$95$1 - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 695.0], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + (-t)), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -720

   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. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. associate-+r+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. +-lowering-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. log-lowering-log.f64 N/A

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

      11. log-lowering-log.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. log-lowering-log.f64 N/A

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

      14. +-lowering-+.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 91.8

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

   7. Simplified 91.8%

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

   if -720 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 695

   1. Initial program 99.4%

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

      1. +-commutative N/A

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

      2. associate-+r- N/A

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

      3. --lowering--.f64 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. +-lowering-+.f64 N/A

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

      7. metadata-eval N/A

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

      8. log-lowering-log.f64 N/A

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

      9. sum-log N/A

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

      10. log-lowering-log.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-lowering-+.f64 99.6

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

   4. Applied egg-rr 99.6%

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

   if 695 < (+.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 x around 0

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. log-lowering-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-lowering-+.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. log-lowering-log.f64 74.6

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

   5. Simplified 74.6%

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

   6. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 61.7

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

   8. Simplified 61.7%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(x + y\right) \leq -720:\\ \;\;\;\;\left(\log \left(x + y\right) - t\right) + a \cdot \log t\\ \mathbf{elif}\;\log z + \log \left(x + y\right) \leq 695:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, \log \left(z \cdot \left(x + y\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right)\\ t_2 := \log z + t\_1\\ \mathbf{if}\;t\_2 \leq -720:\\ \;\;\;\;\left(t\_1 - t\right) + a \cdot \log t\\ \mathbf{elif}\;t\_2 \leq 695:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ x y))) (t_2 (+ (log z) t_1)))
   (if (<= t_2 -720.0)
     (+ (- t_1 t) (* a (log t)))
     (if (<= t_2 695.0)
       (- (fma (log t) (+ a -0.5) (log (* z y))) t)
       (+ (log y) (fma (log t) (+ a -0.5) (- t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y));
	double t_2 = log(z) + t_1;
	double tmp;
	if (t_2 <= -720.0) {
		tmp = (t_1 - t) + (a * log(t));
	} else if (t_2 <= 695.0) {
		tmp = fma(log(t), (a + -0.5), log((z * y))) - t;
	} else {
		tmp = log(y) + fma(log(t), (a + -0.5), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	t_2 = Float64(log(z) + t_1)
	tmp = 0.0
	if (t_2 <= -720.0)
		tmp = Float64(Float64(t_1 - t) + Float64(a * log(t)));
	elseif (t_2 <= 695.0)
		tmp = Float64(fma(log(t), Float64(a + -0.5), log(Float64(z * y))) - t);
	else
		tmp = Float64(log(y) + fma(log(t), Float64(a + -0.5), Float64(-t)));
	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[(N[Log[z], $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -720.0], N[(N[(t$95$1 - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 695.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + (-t)), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -720

   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. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. associate-+r+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. +-lowering-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. log-lowering-log.f64 N/A

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

      11. log-lowering-log.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. log-lowering-log.f64 N/A

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

      14. +-lowering-+.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 91.8

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

   7. Simplified 91.8%

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

   if -720 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 695

   1. Initial program 99.4%

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

      1. flip-+ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. difference-of-squares N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sum-log N/A

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

      6. log-lowering-log.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. diff-log N/A

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

      10. log-lowering-log.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. +-lowering-+.f64 N/A

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

      13. diff-log N/A

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

      14. log-lowering-log.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. +-lowering-+.f64 64.8

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

   4. Applied egg-rr 64.8%

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

   5. Taylor expanded in x around 0

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

      1. --lowering--.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. log-lowering-log.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 67.9

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

   7. Simplified 67.9%

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

   if 695 < (+.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 x around 0

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. log-lowering-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-lowering-+.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. log-lowering-log.f64 74.6

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

   5. Simplified 74.6%

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

   6. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 61.7

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

   8. Simplified 61.7%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(x + y\right) \leq -720:\\ \;\;\;\;\left(\log \left(x + y\right) - t\right) + a \cdot \log t\\ \mathbf{elif}\;\log z + \log \left(x + y\right) \leq 695:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\log z + \log \left(x + y\right)\right) - t\right) + \log t \cdot \left(a - 0.5\right) \leq -1:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (+ (- (+ (log z) (log (+ x y))) t) (* (log t) (- a 0.5))) -1.0)
   (- (* a (log t)) t)
   (fma (+ a -0.5) (log t) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((((log(z) + log((x + y))) - t) + (log(t) * (a - 0.5))) <= -1.0) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = fma((a + -0.5), log(t), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(Float64(log(z) + log(Float64(x + y))) - t) + Float64(log(t) * Float64(a - 0.5))) <= -1.0)
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = fma(Float64(a + -0.5), log(t), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(N[(N[Log[z], $MachinePrecision] + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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. flip-+ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. difference-of-squares N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sum-log N/A

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

      6. log-lowering-log.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. diff-log N/A

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

      10. log-lowering-log.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. +-lowering-+.f64 N/A

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

      13. diff-log N/A

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

      14. log-lowering-log.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. +-lowering-+.f64 50.5

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

   4. Applied egg-rr 50.5%

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

   5. Taylor expanded in x around 0

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

      1. --lowering--.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. log-lowering-log.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 63.9

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

   7. Simplified 63.9%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 89.6

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

   10. Simplified 89.6%

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

   if -1 < (+.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 99.3%

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

      1. *-commutative N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

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

      6. log-lowering-log.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. /-lowering-/.f64 N/A

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

      10. sub-neg N/A

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

      11. +-lowering-+.f64 N/A

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

      12. metadata-eval 99.1

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

   4. Applied egg-rr 99.1%

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

   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. neg-lowering-neg.f64 55.4

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

   7. Simplified 55.4%

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

      1. +-commutative N/A

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

      2. associate-/r/ N/A

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

      3. /-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. /-rgt-identity N/A

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

      6. /-rgt-identity N/A

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

      7. unpow1 N/A

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

      8. metadata-eval N/A

         \[\leadsto \left(a + \frac{-1}{2}\right) \cdot \log t + \left(\mathsf{neg}\left({t}^{\color{blue}{\left(4 - 3\right)}}\right)\right) \]

      9. metadata-eval N/A

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

      10. pow-div N/A

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

      11. pow-prod-up N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. associate-*r* N/A

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

      15. pow3 N/A

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

      16. associate-/l/ N/A

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

      17. un-div-inv N/A

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

      18. distribute-frac-neg2 N/A

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

   9. Applied egg-rr 55.0%

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

4. Final simplification 76.5%

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

Alternative 9: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, -t\right)\\ \mathbf{elif}\;a - 0.5 \leq -0.1:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, -0.5, \log z - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(x + y\right) - t\right) + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (- a 0.5) -5e+18)
   (+ (log y) (fma (log t) (+ a -0.5) (- t)))
   (if (<= (- a 0.5) -0.1)
     (+ (log y) (fma (log t) -0.5 (- (log z) t)))
     (+ (- (log (+ x y)) t) (* a (log t))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a - 0.5) <= -5e+18) {
		tmp = log(y) + fma(log(t), (a + -0.5), -t);
	} else if ((a - 0.5) <= -0.1) {
		tmp = log(y) + fma(log(t), -0.5, (log(z) - t));
	} else {
		tmp = (log((x + y)) - t) + (a * log(t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(a - 0.5) <= -5e+18)
		tmp = Float64(log(y) + fma(log(t), Float64(a + -0.5), Float64(-t)));
	elseif (Float64(a - 0.5) <= -0.1)
		tmp = Float64(log(y) + fma(log(t), -0.5, Float64(log(z) - t)));
	else
		tmp = Float64(Float64(log(Float64(x + y)) - t) + Float64(a * log(t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(a - 0.5), $MachinePrecision], -5e+18], N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a - 0.5), $MachinePrecision], -0.1], N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * -0.5 + N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -5e18

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. log-lowering-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-lowering-+.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. log-lowering-log.f64 68.0

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

   5. Simplified 68.0%

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

   6. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 68.0

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

   8. Simplified 68.0%

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

   if -5e18 < (-.f64 a #s(literal 1/2 binary64)) < -0.10000000000000001

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. log-lowering-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-lowering-+.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. log-lowering-log.f64 65.3

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

   5. Simplified 65.3%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 64.3%

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

   if -0.10000000000000001 < (-.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. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. associate-+r+ N/A

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

      5. +-lowering-+.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. +-lowering-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. log-lowering-log.f64 N/A

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

      11. log-lowering-log.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. log-lowering-log.f64 N/A

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

      14. +-lowering-+.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 98.8

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

   7. Simplified 98.8%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, -t\right)\\ \mathbf{elif}\;a - 0.5 \leq -0.1:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, -0.5, \log z - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(x + y\right) - t\right) + a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{-6}:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, \log z\right)\\ \mathbf{elif}\;t \leq 14500:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, -0.5, \log z - t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 3.1e-6)
   (+ (log y) (fma (log t) (+ a -0.5) (log z)))
   (if (<= t 14500.0)
     (+ (log y) (fma (log t) -0.5 (- (log z) t)))
     (- (* a (log t)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 3.1e-6) {
		tmp = log(y) + fma(log(t), (a + -0.5), log(z));
	} else if (t <= 14500.0) {
		tmp = log(y) + fma(log(t), -0.5, (log(z) - t));
	} else {
		tmp = (a * log(t)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 3.1e-6)
		tmp = Float64(log(y) + fma(log(t), Float64(a + -0.5), log(z)));
	elseif (t <= 14500.0)
		tmp = Float64(log(y) + fma(log(t), -0.5, Float64(log(z) - t)));
	else
		tmp = Float64(Float64(a * log(t)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 3.1e-6], N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 14500.0], N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * -0.5 + N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.1e-6

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. log-lowering-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-lowering-+.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. log-lowering-log.f64 63.8

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

   5. Simplified 63.8%

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

   6. Taylor expanded in t around 0

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

      1. +-lowering-+.f64 N/A

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

      2. log-lowering-log.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. log-lowering-log.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-lowering-+.f64 N/A

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

      9. log-lowering-log.f64 63.6

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

   8. Simplified 63.6%

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

   if 3.1e-6 < t < 14500

   1. Initial program 97.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. log-lowering-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-lowering-+.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. log-lowering-log.f64 83.1

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

   5. Simplified 83.1%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 83.1%

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

   if 14500 < t

   1. Initial program 99.9%

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

      1. flip-+ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. difference-of-squares N/A

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

      4. *-lowering-*.f64 N/A

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

      5. sum-log N/A

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

      6. log-lowering-log.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. diff-log N/A

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

      10. log-lowering-log.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. +-lowering-+.f64 N/A

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

      13. diff-log N/A

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

      14. log-lowering-log.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. +-lowering-+.f64 48.8

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

   4. Applied egg-rr 48.8%

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

   5. Taylor expanded in x around 0

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

      1. --lowering--.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. log-lowering-log.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 66.3

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

   7. Simplified 66.3%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 99.0

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

   10. Simplified 99.0%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{-6}:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, \log z\right)\\ \mathbf{elif}\;t \leq 14500:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, -0.5, \log z - t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in x around 0

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

   1. associate--l+ N/A

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

   2. +-lowering-+.f64 N/A

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

   3. log-lowering-log.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. log-lowering-log.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. +-lowering-+.f64 N/A

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

   11. --lowering--.f64 N/A

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

   12. log-lowering-log.f64 71.1

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

5. Simplified 71.1%

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

Alternative 12: 58.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in x around 0

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

   1. associate--l+ N/A

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

   2. +-lowering-+.f64 N/A

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

   3. log-lowering-log.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. log-lowering-log.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. +-lowering-+.f64 N/A

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

   11. --lowering--.f64 N/A

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

   12. log-lowering-log.f64 71.1

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

5. Simplified 71.1%

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

6. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 60.0

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

8. Simplified 60.0%

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

Alternative 13: 58.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -1.75 \cdot 10^{+32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{+15}:\\ \;\;\;\;\log y - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= a -1.75e+32) t_1 (if (<= a 4.2e+15) (- (log y) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if (a <= -1.75e+32) {
		tmp = t_1;
	} else if (a <= 4.2e+15) {
		tmp = log(y) - 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 = a * log(t)
    if (a <= (-1.75d+32)) then
        tmp = t_1
    else if (a <= 4.2d+15) then
        tmp = log(y) - 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 = a * Math.log(t);
	double tmp;
	if (a <= -1.75e+32) {
		tmp = t_1;
	} else if (a <= 4.2e+15) {
		tmp = Math.log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if a <= -1.75e+32:
		tmp = t_1
	elif a <= 4.2e+15:
		tmp = math.log(y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (a <= -1.75e+32)
		tmp = t_1;
	elseif (a <= 4.2e+15)
		tmp = Float64(log(y) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if (a <= -1.75e+32)
		tmp = t_1;
	elseif (a <= 4.2e+15)
		tmp = log(y) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.75e+32], t$95$1, If[LessEqual[a, 4.2e+15], N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.75e32 or 4.2e15 < a

   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. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 74.4

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

   5. Simplified 74.4%

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

   if -1.75e32 < a < 4.2e15

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-lowering-+.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. log-lowering-log.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-lowering-+.f64 N/A

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

      11. --lowering--.f64 N/A

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

      12. log-lowering-log.f64 66.4

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

   5. Simplified 66.4%

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

   6. Taylor expanded in t around inf

      \[\leadsto \log y + \color{blue}{-1 \cdot t} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 42.0

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

   8. Simplified 42.0%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.75 \cdot 10^{+32}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{+15}:\\ \;\;\;\;\log y - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{+14}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 8.5e+14) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 8.5e+14) {
		tmp = a * log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 8.5d+14) then
        tmp = a * log(t)
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 8.5e+14) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 8.5e+14:
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 8.5e+14)
		tmp = Float64(a * log(t));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 8.5e+14)
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 8.5e+14], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 8.5e14

   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 a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 51.1

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

   5. Simplified 51.1%

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

   if 8.5e14 < 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. neg-lowering-neg.f64 75.1

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

   5. Simplified 75.1%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8.5 \cdot 10^{+14}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 77.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in 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. neg-lowering-neg.f64 76.4

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

5. Simplified 76.4%

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

6. Final simplification 76.4%

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

Alternative 16: 38.8% 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 99.5%

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

3. Taylor expanded in 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. neg-lowering-neg.f64 35.9

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

5. Simplified 35.9%

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

Alternative 17: 2.4% accurate, 321.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in 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. neg-lowering-neg.f64 35.9

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

5. Simplified 35.9%

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

   1. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - t} \]

   2. flip-- N/A

      \[\leadsto \color{blue}{\frac{0 \cdot 0 - t \cdot t}{0 + t}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{0 \cdot 0 - t \cdot t}{0 + t}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} - t \cdot t}{0 + t} \]

   5. --lowering--.f64 N/A

      \[\leadsto \frac{\color{blue}{0 - t \cdot t}}{0 + t} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{0 - \color{blue}{t \cdot t}}{0 + t} \]

   7. +-lowering-+.f64 13.5

      \[\leadsto \frac{0 - t \cdot t}{\color{blue}{0 + t}} \]

7. Applied egg-rr 13.5%

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

   1. +-lft-identity N/A

      \[\leadsto \frac{0 - t \cdot t}{\color{blue}{t}} \]

   2. flip-- N/A

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot 0 - \left(t \cdot t\right) \cdot \left(t \cdot t\right)}{0 + t \cdot t}}}{t} \]

   3. +-lft-identity N/A

      \[\leadsto \frac{\frac{0 \cdot 0 - \left(t \cdot t\right) \cdot \left(t \cdot t\right)}{\color{blue}{t \cdot t}}}{t} \]

   4. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left(t \cdot t\right) \cdot \left(t \cdot t\right)}{t \cdot \left(t \cdot t\right)}} \]

   5. cube-mult N/A

      \[\leadsto \frac{0 \cdot 0 - \left(t \cdot t\right) \cdot \left(t \cdot t\right)}{\color{blue}{{t}^{3}}} \]

   6. sqr-pow N/A

      \[\leadsto \frac{0 \cdot 0 - \left(t \cdot t\right) \cdot \left(t \cdot t\right)}{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}}} \]

   7. unpow-prod-down N/A

      \[\leadsto \frac{0 \cdot 0 - \left(t \cdot t\right) \cdot \left(t \cdot t\right)}{\color{blue}{{\left(t \cdot t\right)}^{\left(\frac{3}{2}\right)}}} \]

   8. sqr-neg N/A

      \[\leadsto \frac{0 \cdot 0 - \left(t \cdot t\right) \cdot \left(t \cdot t\right)}{{\color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)}}^{\left(\frac{3}{2}\right)}} \]

   9. unpow-prod-down N/A

      \[\leadsto \frac{0 \cdot 0 - \left(t \cdot t\right) \cdot \left(t \cdot t\right)}{\color{blue}{{\left(\mathsf{neg}\left(t\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(t\right)\right)}^{\left(\frac{3}{2}\right)}}} \]

   10. sqr-pow N/A

       \[\leadsto \frac{0 \cdot 0 - \left(t \cdot t\right) \cdot \left(t \cdot t\right)}{\color{blue}{{\left(\mathsf{neg}\left(t\right)\right)}^{3}}} \]

   11. cube-mult N/A

       \[\leadsto \frac{0 \cdot 0 - \left(t \cdot t\right) \cdot \left(t \cdot t\right)}{\color{blue}{\left(\mathsf{neg}\left(t\right)\right) \cdot \left(\left(\mathsf{neg}\left(t\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)}} \]

   12. sqr-neg N/A

       \[\leadsto \frac{0 \cdot 0 - \left(t \cdot t\right) \cdot \left(t \cdot t\right)}{\left(\mathsf{neg}\left(t\right)\right) \cdot \color{blue}{\left(t \cdot t\right)}} \]

   13. associate-/l/ N/A

       \[\leadsto \color{blue}{\frac{\frac{0 \cdot 0 - \left(t \cdot t\right) \cdot \left(t \cdot t\right)}{t \cdot t}}{\mathsf{neg}\left(t\right)}} \]

   14. +-lft-identity N/A

       \[\leadsto \frac{\frac{0 \cdot 0 - \left(t \cdot t\right) \cdot \left(t \cdot t\right)}{\color{blue}{0 + t \cdot t}}}{\mathsf{neg}\left(t\right)} \]

   15. flip-- N/A

       \[\leadsto \frac{\color{blue}{0 - t \cdot t}}{\mathsf{neg}\left(t\right)} \]

   16. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{0 \cdot 0} - t \cdot t}{\mathsf{neg}\left(t\right)} \]

   17. neg-sub0 N/A

       \[\leadsto \frac{0 \cdot 0 - t \cdot t}{\color{blue}{0 - t}} \]

9. Applied egg-rr 2.6%

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

Developer Target 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024205 
(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))))