Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 17.6s

Alternatives: 18

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: 83.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -700:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, 0 - t\right)\\ \mathbf{elif}\;t\_1 \leq 875:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log t}{\frac{1}{a + -0.5}} - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))))
   (if (<= t_1 -700.0)
     (fma (log t) a (- 0.0 t))
     (if (<= t_1 875.0)
       (fma (log t) (+ a -0.5) (log (* y z)))
       (- (/ (log t) (/ 1.0 (+ a -0.5))) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double tmp;
	if (t_1 <= -700.0) {
		tmp = fma(log(t), a, (0.0 - t));
	} else if (t_1 <= 875.0) {
		tmp = fma(log(t), (a + -0.5), log((y * z)));
	} else {
		tmp = (log(t) / (1.0 / (a + -0.5))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (t_1 <= -700.0)
		tmp = fma(log(t), a, Float64(0.0 - t));
	elseif (t_1 <= 875.0)
		tmp = fma(log(t), Float64(a + -0.5), log(Float64(y * 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_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -700.0], N[(N[Log[t], $MachinePrecision] * a + N[(0.0 - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 875.0], N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[Log[N[(y * 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 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))) < -700

   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-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      3. --lowering--.f64 96.5

         \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   5. Simplified 96.5%

      \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   6. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. neg-lowering-neg.f64 96.6

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

   8. Simplified 96.6%

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

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(\log t, \color{blue}{a}, \mathsf{neg}\left(t\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 96.6%

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

   if -700 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 875

   1. Initial program 99.0%

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

      1. 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 94.1%

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

   5. Taylor expanded in t around 0

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

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

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

      2. +-commutative N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

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

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

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

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 91.6

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

   7. Simplified 91.6%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 51.4

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

   10. Simplified 51.4%

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

   if 875 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.6%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Add Preprocessing
   3. 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 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - t\right)} + \frac{\log t}{\frac{1}{a + \frac{-1}{2}}} \]

      3. --lowering--.f64 80.9

         \[\leadsto \color{blue}{\left(0 - t\right)} + \frac{\log t}{\frac{1}{a + -0.5}} \]

   7. Simplified 80.9%

      \[\leadsto \color{blue}{\left(0 - t\right)} + \frac{\log t}{\frac{1}{a + -0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -700:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, 0 - t\right)\\ \mathbf{elif}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq 875:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log t}{\frac{1}{a + -0.5}} - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -700:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, 0 - t\right)\\ \mathbf{elif}\;t\_1 \leq 875:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(\left(x + y\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log t}{\frac{1}{a + -0.5}} - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))))
   (if (<= t_1 -700.0)
     (fma (log t) a (- 0.0 t))
     (if (<= t_1 875.0)
       (fma (log t) -0.5 (log (* (+ x y) z)))
       (- (/ (log t) (/ 1.0 (+ a -0.5))) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double tmp;
	if (t_1 <= -700.0) {
		tmp = fma(log(t), a, (0.0 - t));
	} else if (t_1 <= 875.0) {
		tmp = fma(log(t), -0.5, log(((x + y) * z)));
	} else {
		tmp = (log(t) / (1.0 / (a + -0.5))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (t_1 <= -700.0)
		tmp = fma(log(t), a, Float64(0.0 - t));
	elseif (t_1 <= 875.0)
		tmp = fma(log(t), -0.5, log(Float64(Float64(x + y) * 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_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -700.0], N[(N[Log[t], $MachinePrecision] * a + N[(0.0 - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 875.0], N[(N[Log[t], $MachinePrecision] * -0.5 + N[Log[N[(N[(x + y), $MachinePrecision] * 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 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))) < -700

   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-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      3. --lowering--.f64 96.5

         \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   5. Simplified 96.5%

      \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   6. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. neg-lowering-neg.f64 96.6

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

   8. Simplified 96.6%

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

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(\log t, \color{blue}{a}, \mathsf{neg}\left(t\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 96.6%

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

   if -700 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 875

   1. Initial program 99.0%

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

      1. 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 94.1%

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

   5. Taylor expanded in t around 0

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

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

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

      2. +-commutative N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

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

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

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

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 91.6

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

   7. Simplified 91.6%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 89.6

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

   10. Simplified 89.6%

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

   if 875 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.6%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Add Preprocessing
   3. 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 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - t\right)} + \frac{\log t}{\frac{1}{a + \frac{-1}{2}}} \]

      3. --lowering--.f64 80.9

         \[\leadsto \color{blue}{\left(0 - t\right)} + \frac{\log t}{\frac{1}{a + -0.5}} \]

   7. Simplified 80.9%

      \[\leadsto \color{blue}{\left(0 - t\right)} + \frac{\log t}{\frac{1}{a + -0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -700:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, 0 - t\right)\\ \mathbf{elif}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq 875:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(\left(x + y\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log t}{\frac{1}{a + -0.5}} - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -700:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, 0 - t\right)\\ \mathbf{elif}\;t\_1 \leq 875:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log t}{\frac{1}{a + -0.5}} - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))))
   (if (<= t_1 -700.0)
     (fma (log t) a (- 0.0 t))
     (if (<= t_1 875.0)
       (fma (log t) -0.5 (log (* y z)))
       (- (/ (log t) (/ 1.0 (+ a -0.5))) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double tmp;
	if (t_1 <= -700.0) {
		tmp = fma(log(t), a, (0.0 - t));
	} else if (t_1 <= 875.0) {
		tmp = fma(log(t), -0.5, log((y * z)));
	} else {
		tmp = (log(t) / (1.0 / (a + -0.5))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (t_1 <= -700.0)
		tmp = fma(log(t), a, Float64(0.0 - t));
	elseif (t_1 <= 875.0)
		tmp = fma(log(t), -0.5, log(Float64(y * 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_] := Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, -700.0], N[(N[Log[t], $MachinePrecision] * a + N[(0.0 - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 875.0], N[(N[Log[t], $MachinePrecision] * -0.5 + N[Log[N[(y * 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 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))) < -700

   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-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      3. --lowering--.f64 96.5

         \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   5. Simplified 96.5%

      \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   6. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. neg-lowering-neg.f64 96.6

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

   8. Simplified 96.6%

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

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(\log t, \color{blue}{a}, \mathsf{neg}\left(t\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 96.6%

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

   if -700 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 875

   1. Initial program 99.0%

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

      1. 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 94.1%

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

   5. Taylor expanded in t around 0

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

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

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

      2. +-commutative N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

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

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

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

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 91.6

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

   7. Simplified 91.6%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 51.4

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

   10. Simplified 51.4%

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

   11. Taylor expanded in a around 0

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

   13. Simplified 50.0%

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

   if 875 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.6%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Add Preprocessing
   3. 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 99.6

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - t\right)} + \frac{\log t}{\frac{1}{a + \frac{-1}{2}}} \]

      3. --lowering--.f64 80.9

         \[\leadsto \color{blue}{\left(0 - t\right)} + \frac{\log t}{\frac{1}{a + -0.5}} \]

   7. Simplified 80.9%

      \[\leadsto \color{blue}{\left(0 - t\right)} + \frac{\log t}{\frac{1}{a + -0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -700:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, 0 - t\right)\\ \mathbf{elif}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq 875:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log t}{\frac{1}{a + -0.5}} - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -50:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, 0 - t\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))) -50.0)
   (fma (log t) a (- 0.0 t))
   (* (log t) (+ a -0.5))))

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))) <= -50.0) {
		tmp = fma(log(t), a, (0.0 - t));
	} else {
		tmp = log(t) * (a + -0.5);
	}
	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))) <= -50.0)
		tmp = fma(log(t), a, Float64(0.0 - t));
	else
		tmp = Float64(log(t) * Float64(a + -0.5));
	end
	return 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], -50.0], N[(N[Log[t], $MachinePrecision] * a + N[(0.0 - t), $MachinePrecision]), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $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))) < -50

   1. Initial program 99.7%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - \frac{1}{2}\right) \cdot \log t \]
   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-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      3. --lowering--.f64 85.6

         \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   5. Simplified 85.6%

      \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   6. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. neg-lowering-neg.f64 85.7

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

   8. Simplified 85.7%

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

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(\log t, \color{blue}{a}, \mathsf{neg}\left(t\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 86.3%

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

   if -50 < (+.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 99.4%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      3. --lowering--.f64 53.0

         \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   5. Simplified 53.0%

      \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   6. Taylor expanded in t around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 52.7

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

   8. Simplified 52.7%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -50:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, 0 - t\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq 700:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, \log \left(\left(x + y\right) \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (+ (log (+ x y)) (log z)) 700.0)
   (- (fma (+ a -0.5) (log t) (log (* (+ x y) z))) t)
   (- (* (log t) (+ a -0.5)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((log((x + y)) + log(z)) <= 700.0) {
		tmp = fma((a + -0.5), log(t), log(((x + y) * z))) - t;
	} else {
		tmp = (log(t) * (a + -0.5)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(log(Float64(x + y)) + log(z)) <= 700.0)
		tmp = Float64(fma(Float64(a + -0.5), log(t), log(Float64(Float64(x + y) * z))) - t);
	else
		tmp = Float64(Float64(log(t) * Float64(a + -0.5)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision], 700.0], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 700

   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. associate-+r- N/A

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

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

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

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

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

      5. sub-neg N/A

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

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

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

      7. metadata-eval N/A

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

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

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

      9. sum-log N/A

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

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

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

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

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

      12. +-lowering-+.f64 97.1

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

   4. Applied egg-rr 97.1%

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

   if 700 < (+.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-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      3. --lowering--.f64 76.8

         \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   5. Simplified 76.8%

      \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   6. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. neg-lowering-neg.f64 76.8

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

   8. Simplified 76.8%

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

      1. unsub-neg N/A

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

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

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

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

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

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

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

      5. +-lowering-+.f64 76.8

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

   10. Applied egg-rr 76.8%

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

Alternative 7: 65.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq 700:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (+ (log (+ x y)) (log z)) 700.0)
   (- (fma (log t) (+ a -0.5) (log (* y z))) t)
   (- (* (log t) (+ a -0.5)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((log((x + y)) + log(z)) <= 700.0) {
		tmp = fma(log(t), (a + -0.5), log((y * z))) - t;
	} else {
		tmp = (log(t) * (a + -0.5)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(log(Float64(x + y)) + log(z)) <= 700.0)
		tmp = Float64(fma(log(t), Float64(a + -0.5), log(Float64(y * z))) - t);
	else
		tmp = Float64(Float64(log(t) * Float64(a + -0.5)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision], 700.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 700

   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 97.0%

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

   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 58.4

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

   7. Simplified 58.4%

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

   if 700 < (+.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-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      3. --lowering--.f64 76.8

         \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   5. Simplified 76.8%

      \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   6. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. neg-lowering-neg.f64 76.8

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

   8. Simplified 76.8%

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

      1. unsub-neg N/A

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

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

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

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

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

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

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

      5. +-lowering-+.f64 76.8

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

   10. Applied egg-rr 76.8%

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

Alternative 8: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 430:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log z + \log y\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-1 + \left(\frac{\log \left(x + y\right)}{t} + \frac{a \cdot \log t}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 430.0)
   (fma (log t) (+ a -0.5) (+ (log z) (log y)))
   (* t (+ -1.0 (+ (/ (log (+ x y)) t) (/ (* a (log t)) t))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 430.0) {
		tmp = fma(log(t), (a + -0.5), (log(z) + log(y)));
	} else {
		tmp = t * (-1.0 + ((log((x + y)) / t) + ((a * log(t)) / t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 430.0)
		tmp = fma(log(t), Float64(a + -0.5), Float64(log(z) + log(y)));
	else
		tmp = Float64(t * Float64(-1.0 + Float64(Float64(log(Float64(x + y)) / t) + Float64(Float64(a * log(t)) / t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 430.0], N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(-1.0 + N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision] + N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 430

   1. Initial program 99.4%

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

      1. 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 79.1%

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

   5. Taylor expanded in t around 0

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

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

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

      2. +-commutative N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

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

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

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

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 78.0

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

   7. Simplified 78.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 45.8

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

   10. Simplified 45.8%

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

       1. log-prod N/A

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

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

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

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

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

       4. log-lowering-log.f64 58.1

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

   12. Applied egg-rr 58.1%

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

   if 430 < 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. 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. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

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

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

   5. Simplified 99.8%

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

   6. Taylor expanded in a around inf

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

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

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

      2. *-commutative N/A

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

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

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

      4. log-lowering-log.f64 97.8

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

   8. Simplified 97.8%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 430:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log z + \log y\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-1 + \left(\frac{\log \left(x + y\right)}{t} + \frac{a \cdot \log t}{t}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 490:\\ \;\;\;\;\log z + \mathsf{fma}\left(\log t, a + -0.5, \log y\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-1 + \left(\frac{\log \left(x + y\right)}{t} + \frac{a \cdot \log t}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 490.0)
   (+ (log z) (fma (log t) (+ a -0.5) (log y)))
   (* t (+ -1.0 (+ (/ (log (+ x y)) t) (/ (* a (log t)) t))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 490.0) {
		tmp = log(z) + fma(log(t), (a + -0.5), log(y));
	} else {
		tmp = t * (-1.0 + ((log((x + y)) / t) + ((a * log(t)) / t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 490.0)
		tmp = Float64(log(z) + fma(log(t), Float64(a + -0.5), log(y)));
	else
		tmp = Float64(t * Float64(-1.0 + Float64(Float64(log(Float64(x + y)) / t) + Float64(Float64(a * log(t)) / t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 490.0], N[(N[Log[z], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(-1.0 + N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision] + N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 490

   1. Initial program 99.4%

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

      1. 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 79.1%

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

   5. Taylor expanded in t around 0

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

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

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

      2. +-commutative N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

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

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

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

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 78.0

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

   7. Simplified 78.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 45.8

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

   10. Simplified 45.8%

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

   11. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. remove-double-neg N/A

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

       3. log-rec N/A

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

       4. mul-1-neg N/A

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

       5. associate-+r+ N/A

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

       6. +-commutative N/A

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

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

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

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

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

       9. +-commutative N/A

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

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

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

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

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

       12. sub-neg N/A

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

       13. metadata-eval N/A

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

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

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

       15. mul-1-neg N/A

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

       16. log-rec N/A

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

       17. remove-double-neg N/A

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

       18. log-lowering-log.f64 58.0

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

   13. Simplified 58.0%

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

   if 490 < 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. 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. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

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

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

   5. Simplified 99.8%

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

   6. Taylor expanded in a around inf

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

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

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

      2. *-commutative N/A

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

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

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

      4. log-lowering-log.f64 97.8

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

   8. Simplified 97.8%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 490:\\ \;\;\;\;\log z + \mathsf{fma}\left(\log t, a + -0.5, \log y\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-1 + \left(\frac{\log \left(x + y\right)}{t} + \frac{a \cdot \log t}{t}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-+r+ N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

   8. metadata-eval N/A

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

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

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

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

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

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

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

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

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

   13. log-lowering-log.f64 99.5

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

4. Applied egg-rr 99.5%

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

Alternative 11: 68.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0

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

   1. associate--l+ N/A

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

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

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

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

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

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

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

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

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

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

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

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

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

   12. log-lowering-log.f64 62.7

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

5. Simplified 62.7%

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

Alternative 12: 65.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -4 \cdot 10^{+18}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a - 0.5 \leq -0.4:\\ \;\;\;\;\log t \cdot -0.5 - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (- a 0.5) -4e+18)
   (* a (log t))
   (if (<= (- a 0.5) -0.4) (- (* (log t) -0.5) t) (* (log t) (+ a -0.5)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a - 0.5) <= -4e+18) {
		tmp = a * log(t);
	} else if ((a - 0.5) <= -0.4) {
		tmp = (log(t) * -0.5) - t;
	} else {
		tmp = log(t) * (a + -0.5);
	}
	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 ((a - 0.5d0) <= (-4d+18)) then
        tmp = a * log(t)
    else if ((a - 0.5d0) <= (-0.4d0)) then
        tmp = (log(t) * (-0.5d0)) - t
    else
        tmp = log(t) * (a + (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a - 0.5) <= -4e+18) {
		tmp = a * Math.log(t);
	} else if ((a - 0.5) <= -0.4) {
		tmp = (Math.log(t) * -0.5) - t;
	} else {
		tmp = Math.log(t) * (a + -0.5);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a - 0.5) <= -4e+18:
		tmp = a * math.log(t)
	elif (a - 0.5) <= -0.4:
		tmp = (math.log(t) * -0.5) - t
	else:
		tmp = math.log(t) * (a + -0.5)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(a - 0.5) <= -4e+18)
		tmp = Float64(a * log(t));
	elseif (Float64(a - 0.5) <= -0.4)
		tmp = Float64(Float64(log(t) * -0.5) - t);
	else
		tmp = Float64(log(t) * Float64(a + -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a - 0.5) <= -4e+18)
		tmp = a * log(t);
	elseif ((a - 0.5) <= -0.4)
		tmp = (log(t) * -0.5) - t;
	else
		tmp = log(t) * (a + -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(a - 0.5), $MachinePrecision], -4e+18], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4], N[(N[(N[Log[t], $MachinePrecision] * -0.5), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -4e18

   1. Initial program 99.7%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

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

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

      3. log-lowering-log.f64 86.0

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

   5. Simplified 86.0%

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

   if -4e18 < (-.f64 a #s(literal 1/2 binary64)) < -0.40000000000000002

   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-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      3. --lowering--.f64 50.6

         \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   5. Simplified 50.6%

      \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   6. Taylor expanded in a around 0

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

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

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \log t - t} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\log t \cdot \frac{-1}{2}} - t \]

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

         \[\leadsto \color{blue}{\log t \cdot \frac{-1}{2}} - t \]

      4. log-lowering-log.f64 50.3

         \[\leadsto \color{blue}{\log t} \cdot -0.5 - t \]

   8. Simplified 50.3%

      \[\leadsto \color{blue}{\log t \cdot -0.5 - t} \]

   if -0.40000000000000002 < (-.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 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-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      3. --lowering--.f64 98.2

         \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   5. Simplified 98.2%

      \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   6. Taylor expanded in t around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 80.0

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

   8. Simplified 80.0%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -4 \cdot 10^{+18}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a - 0.5 \leq -0.4:\\ \;\;\;\;\log t \cdot -0.5 - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -15200000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2300000:\\ \;\;\;\;0 - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= a -15200000000.0) t_1 (if (<= a 2300000.0) (- 0.0 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 <= -15200000000.0) {
		tmp = t_1;
	} else if (a <= 2300000.0) {
		tmp = 0.0 - 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 <= (-15200000000.0d0)) then
        tmp = t_1
    else if (a <= 2300000.0d0) then
        tmp = 0.0d0 - 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 <= -15200000000.0) {
		tmp = t_1;
	} else if (a <= 2300000.0) {
		tmp = 0.0 - 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 <= -15200000000.0:
		tmp = t_1
	elif a <= 2300000.0:
		tmp = 0.0 - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (a <= -15200000000.0)
		tmp = t_1;
	elseif (a <= 2300000.0)
		tmp = Float64(0.0 - 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 <= -15200000000.0)
		tmp = t_1;
	elseif (a <= 2300000.0)
		tmp = 0.0 - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -15200000000.0], t$95$1, If[LessEqual[a, 2300000.0], N[(0.0 - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.52e10 or 2.3e6 < a

   1. Initial program 99.7%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

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

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

      3. log-lowering-log.f64 82.9

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

   5. Simplified 82.9%

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

   if -1.52e10 < a < 2.3e6

   1. Initial program 99.5%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 44.6

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

   5. Simplified 44.6%

      \[\leadsto \color{blue}{0 - t} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 44.6

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

   7. Applied egg-rr 44.6%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -15200000000:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a \leq 2300000:\\ \;\;\;\;0 - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 72000000000000:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 72000000000000.0) (* (log t) (+ a -0.5)) (- 0.0 t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 72000000000000.0) {
		tmp = log(t) * (a + -0.5);
	} else {
		tmp = 0.0 - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 72000000000000.0d0) then
        tmp = log(t) * (a + (-0.5d0))
    else
        tmp = 0.0d0 - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 72000000000000.0) {
		tmp = Math.log(t) * (a + -0.5);
	} else {
		tmp = 0.0 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 72000000000000.0:
		tmp = math.log(t) * (a + -0.5)
	else:
		tmp = 0.0 - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 72000000000000.0)
		tmp = Float64(log(t) * Float64(a + -0.5));
	else
		tmp = Float64(0.0 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 72000000000000.0)
		tmp = log(t) * (a + -0.5);
	else
		tmp = 0.0 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 72000000000000.0], N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision], N[(0.0 - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.2e13

   1. Initial program 99.4%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

      3. --lowering--.f64 55.5

         \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   5. Simplified 55.5%

      \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

   6. Taylor expanded in t around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 54.3

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

   8. Simplified 54.3%

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

   if 7.2e13 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 74.3

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

   5. Simplified 74.3%

      \[\leadsto \color{blue}{0 - t} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 74.3

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

   7. Applied egg-rr 74.3%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 72000000000000:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0 - t\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 77.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\log t, a + -0.5, 0 - t\right) \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (fma (log t) (+ a -0.5) (- 0.0 t)))

C

double code(double x, double y, double z, double t, double a) {
	return fma(log(t), (a + -0.5), (0.0 - t));
}

Julia

function code(x, y, z, t, a)
	return fma(log(t), Float64(a + -0.5), Float64(0.0 - t))
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[(0.0 - t), $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. 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-sub0 N/A

      \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

   3. --lowering--.f64 73.0

      \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

5. Simplified 73.0%

   \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

6. Taylor expanded in t around 0

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

   1. mul-1-neg N/A

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

   2. +-commutative N/A

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

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

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

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

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

   8. neg-lowering-neg.f64 73.0

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

8. Simplified 73.0%

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

9. Final simplification 73.0%

   \[\leadsto \mathsf{fma}\left(\log t, a + -0.5, 0 - t\right) \]
10. Add Preprocessing

Alternative 16: 77.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
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-sub0 N/A

      \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - \frac{1}{2}\right) \cdot \log t \]

   3. --lowering--.f64 73.0

      \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

5. Simplified 73.0%

   \[\leadsto \color{blue}{\left(0 - t\right)} + \left(a - 0.5\right) \cdot \log t \]

6. Taylor expanded in t around 0

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

   1. mul-1-neg N/A

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

   2. +-commutative N/A

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

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

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

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

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

   8. neg-lowering-neg.f64 73.0

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

8. Simplified 73.0%

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

   1. unsub-neg N/A

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

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

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

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

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

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

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

   5. +-lowering-+.f64 73.0

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

10. Applied egg-rr 73.0%

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

Alternative 17: 37.6% accurate, 80.3× speedup

Math

\[\begin{array}{l} \\ 0 - t \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (- 0.0 t))

C

double code(double x, double y, double z, double t, double a) {
	return 0.0 - 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 = 0.0d0 - t
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return 0.0 - t;
}

Python

def code(x, y, z, t, a):
	return 0.0 - t

Julia

function code(x, y, z, t, a)
	return Float64(0.0 - t)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = 0.0 - t;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(0.0 - 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

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

   3. --lowering--.f64 32.0

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

5. Simplified 32.0%

   \[\leadsto \color{blue}{0 - t} \]
6. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 32.0

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

7. Applied egg-rr 32.0%

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

8. Final simplification 32.0%

   \[\leadsto 0 - t \]
9. Add Preprocessing

Alternative 18: 2.4% accurate, 321.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
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-sub0 N/A

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

   3. --lowering--.f64 32.0

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

5. Simplified 32.0%

   \[\leadsto \color{blue}{0 - t} \]
6. Step-by-step derivation

   1. flip-- N/A

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

   2. metadata-eval N/A

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

   3. neg-sub0 N/A

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

   4. +-lft-identity N/A

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

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

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

   6. neg-sub0 N/A

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

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

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

   8. +-lft-identity N/A

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

   9. +-commutative N/A

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

   10. distribute-rgt-out N/A

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

   11. mul0-lft N/A

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

   12. accelerator-lowering-fma.f64 17.2

       \[\leadsto \frac{0 - \color{blue}{\mathsf{fma}\left(t, t, 0\right)}}{t} \]

7. Applied egg-rr 17.2%

   \[\leadsto \color{blue}{\frac{0 - \mathsf{fma}\left(t, t, 0\right)}{t}} \]
8. Step-by-step derivation

   1. flip3-- N/A

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

   2. metadata-eval N/A

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

   3. +-lft-identity N/A

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

   4. distribute-rgt-in N/A

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

   5. +-rgt-identity N/A

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

   6. associate-/l/ N/A

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

9. Applied egg-rr 2.6%

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

Developer Target 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024196 
(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))))