Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 16.5s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

Alternative 2: 92.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right)\\ t_2 := \left(\left(t\_1 + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_3 := \mathsf{fma}\left(a + -0.5, \log t, t\_1\right) - t\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+15}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 890:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(\left(x + y\right) \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ x y)))
        (t_2 (+ (- (+ t_1 (log z)) t) (* (- a 0.5) (log t))))
        (t_3 (- (fma (+ a -0.5) (log t) t_1) t)))
   (if (<= t_2 -2e+15)
     t_3
     (if (<= t_2 890.0) (- (fma -0.5 (log t) (log (* (+ x y) z))) t) t_3))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y));
	double t_2 = ((t_1 + log(z)) - t) + ((a - 0.5) * log(t));
	double t_3 = fma((a + -0.5), log(t), t_1) - t;
	double tmp;
	if (t_2 <= -2e+15) {
		tmp = t_3;
	} else if (t_2 <= 890.0) {
		tmp = fma(-0.5, log(t), log(((x + y) * z))) - t;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	t_2 = Float64(Float64(Float64(t_1 + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_3 = Float64(fma(Float64(a + -0.5), log(t), t_1) - t)
	tmp = 0.0
	if (t_2 <= -2e+15)
		tmp = t_3;
	elseif (t_2 <= 890.0)
		tmp = Float64(fma(-0.5, log(t), log(Float64(Float64(x + y) * z))) - t);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + t$95$1), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+15], t$95$3, If[LessEqual[t$95$2, 890.0], N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$3]]]]]

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))) < -2e15 or 890 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. associate-+r+ N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 93.9

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

   7. Applied rewrites 93.9%

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

   if -2e15 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 890

   1. Initial program 98.9%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift--.f64 N/A

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

      12. flip-- N/A

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

      13. associate-*l/ N/A

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

      14. div-inv N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 93.7%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 89.9

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

   7. Applied rewrites 89.9%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right) - t\\ \mathbf{elif}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq 890:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(\left(x + y\right) \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right)\\ t_2 := \left(\left(t\_1 + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_2 \leq -600:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{elif}\;t\_2 \leq 890:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, t\_1\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ x y)))
        (t_2 (+ (- (+ t_1 (log z)) t) (* (- a 0.5) (log t)))))
   (if (<= t_2 -600.0)
     (+ (- (log z) t) (* a (log t)))
     (if (<= t_2 890.0)
       (fma (log t) (+ a -0.5) (log (* y z)))
       (- (fma (+ a -0.5) (log t) t_1) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y));
	double t_2 = ((t_1 + log(z)) - t) + ((a - 0.5) * log(t));
	double tmp;
	if (t_2 <= -600.0) {
		tmp = (log(z) - t) + (a * log(t));
	} else if (t_2 <= 890.0) {
		tmp = fma(log(t), (a + -0.5), log((y * z)));
	} else {
		tmp = fma((a + -0.5), log(t), t_1) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	t_2 = Float64(Float64(Float64(t_1 + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (t_2 <= -600.0)
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	elseif (t_2 <= 890.0)
		tmp = fma(log(t), Float64(a + -0.5), log(Float64(y * z)));
	else
		tmp = Float64(fma(Float64(a + -0.5), log(t), t_1) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -600.0], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 890.0], N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + t$95$1), $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))) < -600

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. associate-+r+ N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 96.4

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

   7. Applied rewrites 96.4%

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

   if -600 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 890

   1. Initial program 98.9%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip-- N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\log \left(\left(x + y\right) \cdot z\right)}^{2} - t \cdot t, \frac{1}{\log \left(\left(x + y\right) \cdot z\right) + t}, \left(a + -0.5\right) \cdot \log t\right)} \]

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 97.1

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

   7. Applied rewrites 97.1%

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

   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. lower-fma.f64 N/A

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

      3. lower-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. lower-+.f64 N/A

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

      7. lower-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. lower-*.f64 47.1

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

   10. Applied rewrites 47.1%

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

   if 890 < (+.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. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. associate-+r+ N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 82.1

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

   7. Applied rewrites 82.1%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -600:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{elif}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq 890:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\log z - t\right) + a \cdot \log t\\ \mathbf{if}\;t\_1 \leq -600:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 890:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (+ (- (log z) t) (* a (log t)))))
   (if (<= t_1 -600.0)
     t_2
     (if (<= t_1 890.0) (fma (log t) (+ a -0.5) (log (* y z))) t_2))))

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 t_2 = (log(z) - t) + (a * log(t));
	double tmp;
	if (t_1 <= -600.0) {
		tmp = t_2;
	} else if (t_1 <= 890.0) {
		tmp = fma(log(t), (a + -0.5), log((y * z)));
	} else {
		tmp = t_2;
	}
	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)))
	t_2 = Float64(Float64(log(z) - t) + Float64(a * log(t)))
	tmp = 0.0
	if (t_1 <= -600.0)
		tmp = t_2;
	elseif (t_1 <= 890.0)
		tmp = fma(log(t), Float64(a + -0.5), log(Float64(y * z)));
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -600.0], t$95$2, If[LessEqual[t$95$1, 890.0], N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -600 or 890 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. associate-+r+ N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 91.8

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

   7. Applied rewrites 91.8%

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

   if -600 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 890

   1. Initial program 98.9%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip-- N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\log \left(\left(x + y\right) \cdot z\right)}^{2} - t \cdot t, \frac{1}{\log \left(\left(x + y\right) \cdot z\right) + t}, \left(a + -0.5\right) \cdot \log t\right)} \]

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 97.1

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

   7. Applied rewrites 97.1%

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

   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. lower-fma.f64 N/A

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

      3. lower-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. lower-+.f64 N/A

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

      7. lower-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. lower-*.f64 47.1

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

   10. Applied rewrites 47.1%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -600:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{elif}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq 890:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\log z - t\right) + a \cdot \log t\\ \mathbf{if}\;t\_1 \leq -600:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 890:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(\left(x + y\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (+ (- (log z) t) (* a (log t)))))
   (if (<= t_1 -600.0)
     t_2
     (if (<= t_1 890.0) (fma (log t) -0.5 (log (* (+ x y) z))) t_2))))

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 t_2 = (log(z) - t) + (a * log(t));
	double tmp;
	if (t_1 <= -600.0) {
		tmp = t_2;
	} else if (t_1 <= 890.0) {
		tmp = fma(log(t), -0.5, log(((x + y) * z)));
	} else {
		tmp = t_2;
	}
	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)))
	t_2 = Float64(Float64(log(z) - t) + Float64(a * log(t)))
	tmp = 0.0
	if (t_1 <= -600.0)
		tmp = t_2;
	elseif (t_1 <= 890.0)
		tmp = fma(log(t), -0.5, log(Float64(Float64(x + y) * z)));
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -600.0], t$95$2, If[LessEqual[t$95$1, 890.0], N[(N[Log[t], $MachinePrecision] * -0.5 + N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -600 or 890 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. associate-+r+ N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 91.8

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

   7. Applied rewrites 91.8%

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

   if -600 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 890

   1. Initial program 98.9%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip-- N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\log \left(\left(x + y\right) \cdot z\right)}^{2} - t \cdot t, \frac{1}{\log \left(\left(x + y\right) \cdot z\right) + t}, \left(a + -0.5\right) \cdot \log t\right)} \]

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 97.1

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

   7. Applied rewrites 97.1%

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

   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. lower-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. lower-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. lower-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. lower-*.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. lower-+.f64 94.8

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

   10. Applied rewrites 94.8%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -600:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{elif}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq 890:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(\left(x + y\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\log z - t\right) + a \cdot \log t\\ \mathbf{if}\;t\_1 \leq -600:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 890:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (+ (- (log z) t) (* a (log t)))))
   (if (<= t_1 -600.0)
     t_2
     (if (<= t_1 890.0) (fma (log t) -0.5 (log (* y z))) t_2))))

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 t_2 = (log(z) - t) + (a * log(t));
	double tmp;
	if (t_1 <= -600.0) {
		tmp = t_2;
	} else if (t_1 <= 890.0) {
		tmp = fma(log(t), -0.5, log((y * z)));
	} else {
		tmp = t_2;
	}
	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)))
	t_2 = Float64(Float64(log(z) - t) + Float64(a * log(t)))
	tmp = 0.0
	if (t_1 <= -600.0)
		tmp = t_2;
	elseif (t_1 <= 890.0)
		tmp = fma(log(t), -0.5, log(Float64(y * z)));
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -600.0], t$95$2, If[LessEqual[t$95$1, 890.0], N[(N[Log[t], $MachinePrecision] * -0.5 + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -600 or 890 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. associate-+r+ N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 91.8

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

   7. Applied rewrites 91.8%

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

   if -600 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 890

   1. Initial program 98.9%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip-- N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\log \left(\left(x + y\right) \cdot z\right)}^{2} - t \cdot t, \frac{1}{\log \left(\left(x + y\right) \cdot z\right) + t}, \left(a + -0.5\right) \cdot \log t\right)} \]

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 97.1

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

   7. Applied rewrites 97.1%

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

   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. lower-fma.f64 N/A

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

      3. lower-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. lower-+.f64 N/A

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

      7. lower-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. lower-*.f64 47.1

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

   10. Applied rewrites 47.1%

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

   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. Applied rewrites 45.1%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -600:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{elif}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq 890:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right)\\ t_2 := t\_1 + \log z\\ \mathbf{if}\;t\_2 \leq -750:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{elif}\;t\_2 \leq 650:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, \log \left(\left(x + y\right) \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, t\_1\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ x y))) (t_2 (+ t_1 (log z))))
   (if (<= t_2 -750.0)
     (+ (- (log z) t) (* a (log t)))
     (if (<= t_2 650.0)
       (- (fma (+ a -0.5) (log t) (log (* (+ x y) z))) t)
       (- (fma (+ a -0.5) (log t) t_1) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y));
	double t_2 = t_1 + log(z);
	double tmp;
	if (t_2 <= -750.0) {
		tmp = (log(z) - t) + (a * log(t));
	} else if (t_2 <= 650.0) {
		tmp = fma((a + -0.5), log(t), log(((x + y) * z))) - t;
	} else {
		tmp = fma((a + -0.5), log(t), t_1) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	t_2 = Float64(t_1 + log(z))
	tmp = 0.0
	if (t_2 <= -750.0)
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	elseif (t_2 <= 650.0)
		tmp = Float64(fma(Float64(a + -0.5), log(t), log(Float64(Float64(x + y) * z))) - t);
	else
		tmp = Float64(fma(Float64(a + -0.5), log(t), t_1) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -750.0], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 650.0], 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[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + t$95$1), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. associate-+r+ N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 79.1

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

   7. Applied rewrites 79.1%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 650

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. associate-+r- N/A

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

      12. lower--.f64 N/A

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

   4. Applied rewrites 99.7%

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

   if 650 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. associate-+r+ N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 83.6

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

   7. Applied rewrites 83.6%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -750:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{elif}\;\log \left(x + y\right) + \log z \leq 650:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, \log \left(\left(x + y\right) \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right)\\ t_2 := t\_1 + \log z\\ \mathbf{if}\;t\_2 \leq -750:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{elif}\;t\_2 \leq 650:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, t\_1\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ x y))) (t_2 (+ t_1 (log z))))
   (if (<= t_2 -750.0)
     (+ (- (log z) t) (* a (log t)))
     (if (<= t_2 650.0)
       (- (fma (log t) (+ a -0.5) (log (* y z))) t)
       (- (fma (+ a -0.5) (log t) t_1) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y));
	double t_2 = t_1 + log(z);
	double tmp;
	if (t_2 <= -750.0) {
		tmp = (log(z) - t) + (a * log(t));
	} else if (t_2 <= 650.0) {
		tmp = fma(log(t), (a + -0.5), log((y * z))) - t;
	} else {
		tmp = fma((a + -0.5), log(t), t_1) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	t_2 = Float64(t_1 + log(z))
	tmp = 0.0
	if (t_2 <= -750.0)
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	elseif (t_2 <= 650.0)
		tmp = Float64(fma(log(t), Float64(a + -0.5), log(Float64(y * z))) - t);
	else
		tmp = Float64(fma(Float64(a + -0.5), log(t), t_1) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -750.0], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 650.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + t$95$1), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. associate-+r+ N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 79.1

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

   7. Applied rewrites 79.1%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 650

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip-- N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 76.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\log \left(\left(x + y\right) \cdot z\right)}^{2} - t \cdot t, \frac{1}{\log \left(\left(x + y\right) \cdot z\right) + t}, \left(a + -0.5\right) \cdot \log t\right)} \]

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-+.f64 N/A

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

   7. Applied rewrites 89.8%

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

   8. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. associate-/l* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f64 55.3

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

   10. Applied rewrites 55.3%

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

   11. Taylor expanded in t around 0

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

       1. mul-1-neg N/A

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

       2. +-commutative N/A

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

       3. associate-+r+ N/A

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

       4. unsub-neg N/A

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

       5. lower--.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-log.f64 N/A

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

       9. sub-neg N/A

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

       10. metadata-eval N/A

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

       11. lower-+.f64 N/A

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

       12. lower-log.f64 N/A

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

       13. lower-*.f64 60.6

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

   13. Applied rewrites 60.6%

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

   if 650 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. associate-+r+ N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 83.6

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

   7. Applied rewrites 83.6%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -750:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{elif}\;\log \left(x + y\right) + \log z \leq 650:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.0012:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 0.0012)
   (+ (log y) (fma (log t) (+ a -0.5) (log z)))
   (- (fma (+ a -0.5) (log t) (log (+ x y))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.0012) {
		tmp = log(y) + fma(log(t), (a + -0.5), log(z));
	} else {
		tmp = fma((a + -0.5), log(t), log((x + y))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 0.0012)
		tmp = Float64(log(y) + fma(log(t), Float64(a + -0.5), log(z)));
	else
		tmp = Float64(fma(Float64(a + -0.5), log(t), log(Float64(x + y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 0.0012], N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 0.00119999999999999989

   1. Initial program 99.3%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip-- N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 71.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\log \left(\left(x + y\right) \cdot z\right)}^{2} - t \cdot t, \frac{1}{\log \left(\left(x + y\right) \cdot z\right) + t}, \left(a + -0.5\right) \cdot \log t\right)} \]

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 71.9

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

   7. Applied rewrites 71.9%

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

   8. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 N/A

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

      15. lower-log.f64 61.1

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

   10. Applied rewrites 61.1%

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

   if 0.00119999999999999989 < t

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift--.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. associate--l+ N/A

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

      13. associate-+r+ N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.7

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

   7. Applied rewrites 98.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.0012:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right) - t\\ \end{array} \]
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. lift-+.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. +-commutative N/A

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

   10. lift--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. associate--l+ N/A

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

   13. associate-+r+ N/A

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

   14. lower-+.f64 N/A

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

4. Applied rewrites 99.6%

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

Alternative 11: 69.2% 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. lower-+.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. lower-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. lower-fma.f64 N/A

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

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

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

   11. lower--.f64 N/A

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

   12. lower-log.f64 68.8

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

5. Applied rewrites 68.8%

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

Alternative 12: 76.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * 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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. +-commutative N/A

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

   10. lift--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. associate--l+ N/A

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

   13. associate-+r+ N/A

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

   14. lower-+.f64 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-log.f64 78.7

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

7. Applied rewrites 78.7%

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

8. Final simplification 78.7%

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

Alternative 13: 61.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a - 0.5 \leq -5 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a - 0.5 \leq 10^{+70}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= (- a 0.5) -5e+25) t_1 (if (<= (- a 0.5) 1e+70) (- t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if ((a - 0.5) <= -5e+25) {
		tmp = t_1;
	} else if ((a - 0.5) <= 1e+70) {
		tmp = -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if ((a - 0.5d0) <= (-5d+25)) then
        tmp = t_1
    else if ((a - 0.5d0) <= 1d+70) then
        tmp = -t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if ((a - 0.5) <= -5e+25) {
		tmp = t_1;
	} else if ((a - 0.5) <= 1e+70) {
		tmp = -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if (a - 0.5) <= -5e+25:
		tmp = t_1
	elif (a - 0.5) <= 1e+70:
		tmp = -t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (Float64(a - 0.5) <= -5e+25)
		tmp = t_1;
	elseif (Float64(a - 0.5) <= 1e+70)
		tmp = Float64(-t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if ((a - 0.5) <= -5e+25)
		tmp = t_1;
	elseif ((a - 0.5) <= 1e+70)
		tmp = -t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -5e+25], t$95$1, If[LessEqual[N[(a - 0.5), $MachinePrecision], 1e+70], (-t), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -5.00000000000000024e25 or 1.00000000000000007e70 < (-.f64 a #s(literal 1/2 binary64))

   1. Initial program 99.7%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 79.3

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

   5. Applied rewrites 79.3%

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

   if -5.00000000000000024e25 < (-.f64 a #s(literal 1/2 binary64)) < 1.00000000000000007e70

   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. lower-neg.f64 53.0

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

   5. Applied rewrites 53.0%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -5 \cdot 10^{+25}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a - 0.5 \leq 10^{+70}:\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 77.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 78.2

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

5. Applied rewrites 78.2%

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

6. Final simplification 78.2%

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

Alternative 15: 37.6% accurate, 107.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 39.2

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

5. Applied rewrites 39.2%

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

Developer Target 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024219 
(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))))