Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 15.6s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

Alternative 2: 86.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in 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 97.4

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

   5. Simplified 97.4%

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

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

   1. Initial program 99.2%

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

   3. Taylor expanded in t around 0

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

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

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

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

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

      3. +-commutative N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 97.8

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

   5. Simplified 97.8%

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

   6. Taylor expanded in y around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. associate-+r+ N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

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

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

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

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

      15. log-lowering-log.f64 50.4

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

   8. Simplified 50.4%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

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

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

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

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

       5. log-lowering-log.f64 50.0

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

   11. Simplified 50.0%

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

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

   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

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

   5. Simplified 60.3%

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

      1. unsub-neg N/A

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

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

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

   7. Applied egg-rr 60.3%

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

   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 99.4

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

   10. Simplified 99.4%

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

4. Final simplification 85.9%

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

Alternative 3: 90.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + t\_1\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+32}:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;t\_2 \leq 880:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(z \cdot \left(x + y\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) (log t)))
        (t_2 (+ (- (+ (log (+ x y)) (log z)) t) t_1)))
   (if (<= t_2 -1e+32)
     (- (* a (log t)) t)
     (if (<= t_2 880.0)
       (- (fma (log t) -0.5 (log (* z (+ x y)))) t)
       (- t_1 t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * log(t);
	double t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	double tmp;
	if (t_2 <= -1e+32) {
		tmp = (a * log(t)) - t;
	} else if (t_2 <= 880.0) {
		tmp = fma(log(t), -0.5, log((z * (x + y)))) - t;
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a - 0.5) * log(t))
	t_2 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + t_1)
	tmp = 0.0
	if (t_2 <= -1e+32)
		tmp = Float64(Float64(a * log(t)) - t);
	elseif (t_2 <= 880.0)
		tmp = Float64(fma(log(t), -0.5, log(Float64(z * Float64(x + y)))) - t);
	else
		tmp = Float64(t_1 - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+32], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$2, 880.0], N[(N[(N[Log[t], $MachinePrecision] * -0.5 + N[Log[N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(t$95$1 - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -1.00000000000000005e32

   1. Initial program 99.8%

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

   3. Taylor expanded in t around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

   5. Simplified 91.1%

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

      1. unsub-neg N/A

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

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

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

   7. Applied egg-rr 91.1%

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

   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 99.8

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

   10. Simplified 99.8%

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

   if -1.00000000000000005e32 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 880

   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. Step-by-step derivation

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

   4. Applied egg-rr 96.2%

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

   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. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 94.5

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

   7. Simplified 94.5%

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

   if 880 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.5%

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

   3. 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 84.3

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

   5. Simplified 84.3%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -1 \cdot 10^{+32}:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq 880:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(z \cdot \left(x + y\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(a - 0.5\right) \cdot \log t - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + t\_1\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+32}:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;t\_2 \leq 880:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) (log t)))
        (t_2 (+ (- (+ (log (+ x y)) (log z)) t) t_1)))
   (if (<= t_2 -1e+32)
     (- (* a (log t)) t)
     (if (<= t_2 880.0) (- (fma (log t) -0.5 (log (* y z))) t) (- t_1 t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * log(t);
	double t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	double tmp;
	if (t_2 <= -1e+32) {
		tmp = (a * log(t)) - t;
	} else if (t_2 <= 880.0) {
		tmp = fma(log(t), -0.5, log((y * z))) - t;
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a - 0.5) * log(t))
	t_2 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + t_1)
	tmp = 0.0
	if (t_2 <= -1e+32)
		tmp = Float64(Float64(a * log(t)) - t);
	elseif (t_2 <= 880.0)
		tmp = Float64(fma(log(t), -0.5, log(Float64(y * z))) - t);
	else
		tmp = Float64(t_1 - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+32], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$2, 880.0], N[(N[(N[Log[t], $MachinePrecision] * -0.5 + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(t$95$1 - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -1.00000000000000005e32

   1. Initial program 99.8%

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

   3. Taylor expanded in t around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

   5. Simplified 91.1%

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

      1. unsub-neg N/A

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

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

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

   7. Applied egg-rr 91.1%

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

   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 99.8

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

   10. Simplified 99.8%

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

   if -1.00000000000000005e32 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 880

   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. Step-by-step derivation

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

   4. Applied egg-rr 96.2%

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

   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. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 94.5

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

   7. Simplified 94.5%

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

   8. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 52.0

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

   10. Simplified 52.0%

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

   if 880 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.5%

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

   3. 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 84.3

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

   5. Simplified 84.3%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -1 \cdot 10^{+32}:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq 880:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(a - 0.5\right) \cdot \log t - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq -800:\\ \;\;\;\;\frac{\log t}{\frac{1}{a + -0.5}} - t\\ \mathbf{elif}\;t\_1 \leq 660:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, \log \left(z \cdot \left(x + y\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a - 0.5\right) \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (<= t_1 -800.0)
     (- (/ (log t) (/ 1.0 (+ a -0.5))) t)
     (if (<= t_1 660.0)
       (fma (+ a -0.5) (log t) (- (log (* z (+ x y))) t))
       (- (* (- a 0.5) (log t)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double tmp;
	if (t_1 <= -800.0) {
		tmp = (log(t) / (1.0 / (a + -0.5))) - t;
	} else if (t_1 <= 660.0) {
		tmp = fma((a + -0.5), log(t), (log((z * (x + y))) - t));
	} else {
		tmp = ((a - 0.5) * log(t)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	tmp = 0.0
	if (t_1 <= -800.0)
		tmp = Float64(Float64(log(t) / Float64(1.0 / Float64(a + -0.5))) - t);
	elseif (t_1 <= 660.0)
		tmp = fma(Float64(a + -0.5), log(t), Float64(log(Float64(z * Float64(x + y))) - t));
	else
		tmp = Float64(Float64(Float64(a - 0.5) * log(t)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -800.0], N[(N[(N[Log[t], $MachinePrecision] / N[(1.0 / N[(a + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$1, 660.0], 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[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 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. 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 99.1

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

   7. Simplified 99.1%

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

   if -800 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 660

   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. 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 99.2

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

   4. Applied egg-rr 99.2%

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

   if 660 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.8%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   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 84.6

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

   5. Simplified 84.6%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -800:\\ \;\;\;\;\frac{\log t}{\frac{1}{a + -0.5}} - t\\ \mathbf{elif}\;\log \left(x + y\right) + \log z \leq 660:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, \log \left(z \cdot \left(x + y\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a - 0.5\right) \cdot \log t - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq -800:\\ \;\;\;\;\frac{\log t}{\frac{1}{a + -0.5}} - t\\ \mathbf{elif}\;t\_1 \leq 660:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(a - 0.5\right) \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (<= t_1 -800.0)
     (- (/ (log t) (/ 1.0 (+ a -0.5))) t)
     (if (<= t_1 660.0)
       (- (fma (log t) (+ a -0.5) (log (* y z))) t)
       (- (* (- a 0.5) (log t)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double tmp;
	if (t_1 <= -800.0) {
		tmp = (log(t) / (1.0 / (a + -0.5))) - t;
	} else if (t_1 <= 660.0) {
		tmp = fma(log(t), (a + -0.5), log((y * z))) - t;
	} else {
		tmp = ((a - 0.5) * log(t)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	tmp = 0.0
	if (t_1 <= -800.0)
		tmp = Float64(Float64(log(t) / Float64(1.0 / Float64(a + -0.5))) - t);
	elseif (t_1 <= 660.0)
		tmp = Float64(fma(log(t), Float64(a + -0.5), log(Float64(y * z))) - t);
	else
		tmp = Float64(Float64(Float64(a - 0.5) * log(t)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -800.0], N[(N[(N[Log[t], $MachinePrecision] / N[(1.0 / N[(a + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$1, 660.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 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. 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 99.1

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

   7. Simplified 99.1%

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

   if -800 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 660

   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. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

   4. Applied egg-rr 99.0%

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

   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. *-lowering-*.f64 58.8

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

   7. Simplified 58.8%

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

   if 660 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.8%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   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 84.6

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

   5. Simplified 84.6%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -800:\\ \;\;\;\;\frac{\log t}{\frac{1}{a + -0.5}} - t\\ \mathbf{elif}\;\log \left(x + y\right) + \log z \leq 660:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(a - 0.5\right) \cdot \log t - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 40.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t))) <= -2000.0) {
		tmp = -t;
	} else {
		tmp = log(z);
	}
	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 ((((log((x + y)) + log(z)) - t) + ((a - 0.5d0) * log(t))) <= (-2000.0d0)) then
        tmp = -t
    else
        tmp = log(z)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[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], -2000.0], (-t), N[Log[z], $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))) < -2e3

   1. Initial program 99.8%

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

   3. Taylor expanded in 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 61.0

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

   5. Simplified 61.0%

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

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

   1. Initial program 99.3%

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

   3. Taylor expanded in t around 0

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

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

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

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

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

      3. +-commutative N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 98.2

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

   5. Simplified 98.2%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

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

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

      3. log-lowering-log.f64 49.0

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

   8. Simplified 49.0%

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

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\log z} \]
   10. Step-by-step derivation

       1. log-lowering-log.f64 9.4

          \[\leadsto \color{blue}{\log z} \]

   11. Simplified 9.4%

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

Alternative 8: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3000:\\ \;\;\;\;\log z + \mathsf{fma}\left(\log t, a + -0.5, \log \left(x + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log t}{\frac{1}{a + -0.5}} - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 3000.0)
   (+ (log z) (fma (log t) (+ a -0.5) (log (+ x y))))
   (- (/ (log t) (/ 1.0 (+ a -0.5))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 3000.0) {
		tmp = log(z) + fma(log(t), (a + -0.5), log((x + y)));
	} else {
		tmp = (log(t) / (1.0 / (a + -0.5))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 3000.0)
		tmp = Float64(log(z) + fma(log(t), Float64(a + -0.5), log(Float64(x + y))));
	else
		tmp = Float64(Float64(log(t) / Float64(1.0 / Float64(a + -0.5))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 3000.0], N[(N[Log[z], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] / N[(1.0 / N[(a + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3e3

   1. Initial program 99.4%

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

   3. Taylor expanded in t around 0

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

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

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

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

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

      3. +-commutative N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 98.7

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

   5. Simplified 98.7%

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

   if 3e3 < 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. *-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 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 97.5

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

   7. Simplified 97.5%

      \[\leadsto \color{blue}{\left(-t\right)} + \frac{\log t}{\frac{1}{a + -0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3000:\\ \;\;\;\;\log z + \mathsf{fma}\left(\log t, a + -0.5, \log \left(x + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log t}{\frac{1}{a + -0.5}} - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3000:\\ \;\;\;\;\log z + \mathsf{fma}\left(\log t, a + -0.5, \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log t}{\frac{1}{a + -0.5}} - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 3000.0)
   (+ (log z) (fma (log t) (+ a -0.5) (log y)))
   (- (/ (log t) (/ 1.0 (+ a -0.5))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 3000.0) {
		tmp = log(z) + fma(log(t), (a + -0.5), log(y));
	} else {
		tmp = (log(t) / (1.0 / (a + -0.5))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 3000.0)
		tmp = Float64(log(z) + fma(log(t), Float64(a + -0.5), log(y)));
	else
		tmp = Float64(Float64(log(t) / Float64(1.0 / Float64(a + -0.5))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 3000.0], N[(N[Log[z], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] / N[(1.0 / N[(a + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3e3

   1. Initial program 99.4%

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

   3. Taylor expanded in t around 0

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

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

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

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

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

      3. +-commutative N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 98.7

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

   5. Simplified 98.7%

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

   6. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

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

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

      10. log-lowering-log.f64 59.3

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

   8. Simplified 59.3%

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

   if 3e3 < 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. *-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 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 97.5

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

   7. Simplified 97.5%

      \[\leadsto \color{blue}{\left(-t\right)} + \frac{\log t}{\frac{1}{a + -0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3000:\\ \;\;\;\;\log z + \mathsf{fma}\left(\log t, a + -0.5, \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log t}{\frac{1}{a + -0.5}} - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3000:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log t}{\frac{1}{a + -0.5}} - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 3000.0)
   (+ (log y) (fma (log t) (+ a -0.5) (log z)))
   (- (/ (log t) (/ 1.0 (+ a -0.5))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 3000.0) {
		tmp = log(y) + fma(log(t), (a + -0.5), log(z));
	} else {
		tmp = (log(t) / (1.0 / (a + -0.5))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 3000.0)
		tmp = Float64(log(y) + fma(log(t), Float64(a + -0.5), log(z)));
	else
		tmp = Float64(Float64(log(t) / Float64(1.0 / Float64(a + -0.5))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 3000.0], N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] / N[(1.0 / N[(a + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3e3

   1. Initial program 99.4%

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

   3. Taylor expanded in t around 0

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

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

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

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

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

      3. +-commutative N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 98.7

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

   5. Simplified 98.7%

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

   6. Taylor expanded in y around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. associate-+r+ N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

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

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

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

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

      15. log-lowering-log.f64 59.3

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

   8. Simplified 59.3%

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

   if 3e3 < 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. *-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 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 97.5

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

   7. Simplified 97.5%

      \[\leadsto \color{blue}{\left(-t\right)} + \frac{\log t}{\frac{1}{a + -0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3000:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log t}{\frac{1}{a + -0.5}} - t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a - 0.5 \leq -0.5000000000000001:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a - 0.5 \leq 5 \cdot 10^{+26}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= (- a 0.5) -0.5000000000000001)
     t_1
     (if (<= (- a 0.5) 5e+26) (- 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 - 0.5) <= -0.5000000000000001) {
		tmp = t_1;
	} else if ((a - 0.5) <= 5e+26) {
		tmp = -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if ((a - 0.5d0) <= (-0.5000000000000001d0)) then
        tmp = t_1
    else if ((a - 0.5d0) <= 5d+26) then
        tmp = -t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if ((a - 0.5) <= -0.5000000000000001) {
		tmp = t_1;
	} else if ((a - 0.5) <= 5e+26) {
		tmp = -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 - 0.5) <= -0.5000000000000001:
		tmp = t_1
	elif (a - 0.5) <= 5e+26:
		tmp = -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 (Float64(a - 0.5) <= -0.5000000000000001)
		tmp = t_1;
	elseif (Float64(a - 0.5) <= 5e+26)
		tmp = Float64(-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 - 0.5) <= -0.5000000000000001)
		tmp = t_1;
	elseif ((a - 0.5) <= 5e+26)
		tmp = -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[N[(a - 0.5), $MachinePrecision], -0.5000000000000001], t$95$1, If[LessEqual[N[(a - 0.5), $MachinePrecision], 5e+26], (-t), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -0.500000000000000111 or 5.0000000000000001e26 < (-.f64 a #s(literal 1/2 binary64))

   1. Initial program 99.6%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

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

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

      3. log-lowering-log.f64 80.0

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

   5. Simplified 80.0%

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

   if -0.500000000000000111 < (-.f64 a #s(literal 1/2 binary64)) < 5.0000000000000001e26

   1. Initial program 99.6%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 51.3

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

   5. Simplified 51.3%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -0.5000000000000001:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a - 0.5 \leq 5 \cdot 10^{+26}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in 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 76.9

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

5. Simplified 76.9%

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

6. Final simplification 76.9%

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

Alternative 13: 73.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in t around inf

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. mul-1-neg N/A

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

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

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

5. Simplified 87.8%

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

   1. unsub-neg N/A

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

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

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

7. Applied egg-rr 87.9%

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

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 74.3

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

10. Simplified 74.3%

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

11. Final simplification 74.3%

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

Alternative 14: 37.7% accurate, 107.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 36.1

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

5. Simplified 36.1%

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

Alternative 15: 2.5% 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 99.6%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 36.1

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

5. Simplified 36.1%

   \[\leadsto \color{blue}{-t} \]
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 18.0

      \[\leadsto \frac{0 - t \cdot t}{\color{blue}{0 + t}} \]

7. Applied egg-rr 18.0%

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

   1. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0 \cdot 0} - t \cdot t}{0 + t} \]

   2. flip-- N/A

      \[\leadsto \color{blue}{0 - t} \]

   3. neg-sub0 N/A

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

   4. +-lft-identity N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(0 + t\right)}\right) \]

   5. flip3-+ N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{0}^{3} + {t}^{3}}{0 \cdot 0 + \left(t \cdot t - 0 \cdot t\right)}}\right) \]

   6. distribute-neg-frac N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left({0}^{3} + {t}^{3}\right)\right)}{0 \cdot 0 + \left(t \cdot t - 0 \cdot t\right)}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\mathsf{neg}\left(\left(\color{blue}{0} + {t}^{3}\right)\right)}{0 \cdot 0 + \left(t \cdot t - 0 \cdot t\right)} \]

   8. +-lft-identity N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{t}^{3}}\right)}{0 \cdot 0 + \left(t \cdot t - 0 \cdot t\right)} \]

   9. cube-neg N/A

      \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(t\right)\right)}^{3}}}{0 \cdot 0 + \left(t \cdot t - 0 \cdot t\right)} \]

   10. sqr-pow N/A

       \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(t\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(t\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(t \cdot t - 0 \cdot t\right)} \]

   11. unpow-prod-down N/A

       \[\leadsto \frac{\color{blue}{{\left(\left(\mathsf{neg}\left(t\right)\right) \cdot \left(\mathsf{neg}\left(t\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(t \cdot t - 0 \cdot t\right)} \]

   12. sqr-neg N/A

       \[\leadsto \frac{{\color{blue}{\left(t \cdot t\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(t \cdot t - 0 \cdot t\right)} \]

   13. unpow-prod-down N/A

       \[\leadsto \frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(t \cdot t - 0 \cdot t\right)} \]

   14. sqr-pow N/A

       \[\leadsto \frac{\color{blue}{{t}^{3}}}{0 \cdot 0 + \left(t \cdot t - 0 \cdot t\right)} \]

   15. +-lft-identity N/A

       \[\leadsto \frac{\color{blue}{0 + {t}^{3}}}{0 \cdot 0 + \left(t \cdot t - 0 \cdot t\right)} \]

   16. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{{0}^{3}} + {t}^{3}}{0 \cdot 0 + \left(t \cdot t - 0 \cdot t\right)} \]

   17. flip3-+ N/A

       \[\leadsto \color{blue}{0 + t} \]

   18. +-lft-identity 2.2

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

9. Applied egg-rr 2.2%

   \[\leadsto \color{blue}{t} \]
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 2024199 
(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))))