Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 20.4s

Alternatives: 15

Speedup: 0.8×

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, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-+l+ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-def 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. associate-+l- 99.6%

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

   2. associate--l+ 99.6%

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

   3. sub-neg 99.6%

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

   4. +-commutative 99.6%

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

   5. associate--r+ 99.6%

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

   6. sub-neg 99.6%

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

   7. associate-+l- 99.6%

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

   8. neg-sub0 99.6%

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

   9. associate--r+ 99.6%

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

   10. unsub-neg 99.6%

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

   11. associate-+l- 99.6%

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

   12. neg-sub0 99.6%

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

   13. *-commutative 99.6%

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

   14. distribute-rgt-neg-in 99.6%

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

   15. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -40000000000 \lor \neg \left(a - 0.5 \leq -0.2\right):\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - \left(\log t \cdot 0.5 + t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -40000000000.0) (not (<= (- a 0.5) -0.2)))
   (+ (- (log z) t) (* a (log t)))
   (+ (log (+ x y)) (- (log z) (+ (* (log t) 0.5) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -40000000000.0) || !((a - 0.5) <= -0.2)) {
		tmp = (log(z) - t) + (a * log(t));
	} else {
		tmp = log((x + y)) + (log(z) - ((log(t) * 0.5) + t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (((a - 0.5d0) <= (-40000000000.0d0)) .or. (.not. ((a - 0.5d0) <= (-0.2d0)))) then
        tmp = (log(z) - t) + (a * log(t))
    else
        tmp = log((x + y)) + (log(z) - ((log(t) * 0.5d0) + t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -40000000000.0) || !((a - 0.5) <= -0.2)) {
		tmp = (Math.log(z) - t) + (a * Math.log(t));
	} else {
		tmp = Math.log((x + y)) + (Math.log(z) - ((Math.log(t) * 0.5) + t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((a - 0.5) <= -40000000000.0) or not ((a - 0.5) <= -0.2):
		tmp = (math.log(z) - t) + (a * math.log(t))
	else:
		tmp = math.log((x + y)) + (math.log(z) - ((math.log(t) * 0.5) + t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -40000000000.0) || !(Float64(a - 0.5) <= -0.2))
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	else
		tmp = Float64(log(Float64(x + y)) + Float64(log(z) - Float64(Float64(log(t) * 0.5) + t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((a - 0.5) <= -40000000000.0) || ~(((a - 0.5) <= -0.2)))
		tmp = (log(z) - t) + (a * log(t));
	else
		tmp = log((x + y)) + (log(z) - ((log(t) * 0.5) + t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -40000000000.0], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.2]], $MachinePrecision]], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - N[(N[(N[Log[t], $MachinePrecision] * 0.5), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a 1/2) < -4e10 or -0.20000000000000001 < (-.f64 a 1/2)

   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. Step-by-step derivation

      1. associate--l+ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 99.2%

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

      1. *-commutative 99.2%

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

   6. Simplified 99.2%

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

   if -4e10 < (-.f64 a 1/2) < -0.20000000000000001

   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. Step-by-step derivation

      1. associate-+l- 99.5%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. associate--r+ 99.5%

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

      6. sub-neg 99.5%

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

      7. associate-+l- 99.5%

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

      8. neg-sub0 99.5%

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

      9. associate--r+ 99.5%

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

      10. unsub-neg 99.5%

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

      11. associate-+l- 99.5%

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

      12. neg-sub0 99.5%

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

      13. *-commutative 99.5%

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

      14. distribute-rgt-neg-in 99.5%

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

      15. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a around 0 98.7%

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

      1. *-commutative 98.7%

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

   6. Simplified 98.7%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -40000000000 \lor \neg \left(a - 0.5 \leq -0.2\right):\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - \left(\log t \cdot 0.5 + t\right)\right)\\ \end{array} \]

Alternative 4: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -40000000000 \lor \neg \left(a - 0.5 \leq -0.2\right):\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \left(\log y - t\right)\right) + -0.5 \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -40000000000.0) (not (<= (- a 0.5) -0.2)))
   (+ (- (log z) t) (* a (log t)))
   (+ (+ (log z) (- (log y) t)) (* -0.5 (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -40000000000.0) || !((a - 0.5) <= -0.2)) {
		tmp = (log(z) - t) + (a * log(t));
	} else {
		tmp = (log(z) + (log(y) - t)) + (-0.5 * log(t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (((a - 0.5d0) <= (-40000000000.0d0)) .or. (.not. ((a - 0.5d0) <= (-0.2d0)))) then
        tmp = (log(z) - t) + (a * log(t))
    else
        tmp = (log(z) + (log(y) - t)) + ((-0.5d0) * log(t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -40000000000.0) || !((a - 0.5) <= -0.2)) {
		tmp = (Math.log(z) - t) + (a * Math.log(t));
	} else {
		tmp = (Math.log(z) + (Math.log(y) - t)) + (-0.5 * Math.log(t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((a - 0.5) <= -40000000000.0) or not ((a - 0.5) <= -0.2):
		tmp = (math.log(z) - t) + (a * math.log(t))
	else:
		tmp = (math.log(z) + (math.log(y) - t)) + (-0.5 * math.log(t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -40000000000.0) || !(Float64(a - 0.5) <= -0.2))
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	else
		tmp = Float64(Float64(log(z) + Float64(log(y) - t)) + Float64(-0.5 * log(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((a - 0.5) <= -40000000000.0) || ~(((a - 0.5) <= -0.2)))
		tmp = (log(z) - t) + (a * log(t));
	else
		tmp = (log(z) + (log(y) - t)) + (-0.5 * log(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -40000000000.0], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.2]], $MachinePrecision]], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a 1/2) < -4e10 or -0.20000000000000001 < (-.f64 a 1/2)

   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. Step-by-step derivation

      1. associate--l+ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 99.2%

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

      1. *-commutative 99.2%

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

   6. Simplified 99.2%

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

   if -4e10 < (-.f64 a 1/2) < -0.20000000000000001

   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. Step-by-step derivation

      1. associate--l+ 99.5%

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

      2. sub-neg 99.5%

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

      3. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 66.2%

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

      1. associate--l+ 66.2%

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

   6. Simplified 66.2%

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

   7. Taylor expanded in a around 0 66.1%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -40000000000 \lor \neg \left(a - 0.5 \leq -0.2\right):\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \left(\log y - t\right)\right) + -0.5 \cdot \log t\\ \end{array} \]

Alternative 5: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -40000000000 \lor \neg \left(a - 0.5 \leq -0.2\right):\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \left(\log y + -0.5 \cdot \log t\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -40000000000.0) (not (<= (- a 0.5) -0.2)))
   (+ (- (log z) t) (* a (log t)))
   (- (+ (log z) (+ (log y) (* -0.5 (log t)))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -40000000000.0) || !((a - 0.5) <= -0.2)) {
		tmp = (log(z) - t) + (a * log(t));
	} else {
		tmp = (log(z) + (log(y) + (-0.5 * log(t)))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (((a - 0.5d0) <= (-40000000000.0d0)) .or. (.not. ((a - 0.5d0) <= (-0.2d0)))) then
        tmp = (log(z) - t) + (a * log(t))
    else
        tmp = (log(z) + (log(y) + ((-0.5d0) * log(t)))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -40000000000.0) || !((a - 0.5) <= -0.2)) {
		tmp = (Math.log(z) - t) + (a * Math.log(t));
	} else {
		tmp = (Math.log(z) + (Math.log(y) + (-0.5 * Math.log(t)))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((a - 0.5) <= -40000000000.0) or not ((a - 0.5) <= -0.2):
		tmp = (math.log(z) - t) + (a * math.log(t))
	else:
		tmp = (math.log(z) + (math.log(y) + (-0.5 * math.log(t)))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -40000000000.0) || !(Float64(a - 0.5) <= -0.2))
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	else
		tmp = Float64(Float64(log(z) + Float64(log(y) + Float64(-0.5 * log(t)))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((a - 0.5) <= -40000000000.0) || ~(((a - 0.5) <= -0.2)))
		tmp = (log(z) - t) + (a * log(t));
	else
		tmp = (log(z) + (log(y) + (-0.5 * log(t)))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -40000000000.0], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.2]], $MachinePrecision]], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a 1/2) < -4e10 or -0.20000000000000001 < (-.f64 a 1/2)

   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. Step-by-step derivation

      1. associate--l+ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 99.2%

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

      1. *-commutative 99.2%

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

   6. Simplified 99.2%

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

   if -4e10 < (-.f64 a 1/2) < -0.20000000000000001

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. fma-def 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. associate--l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 66.2%

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

   5. Taylor expanded in a around 0 66.2%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -40000000000 \lor \neg \left(a - 0.5 \leq -0.2\right):\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \left(\log y + -0.5 \cdot \log t\right)\right) - t\\ \end{array} \]

Alternative 6: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate--l+ 99.6%

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

   2. sub-neg 99.6%

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

   3. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 7: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -60000000 \lor \neg \left(a \leq 9.5 \cdot 10^{-38}\right):\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log \left(\frac{z}{\sqrt{t}}\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -60000000.0) (not (<= a 9.5e-38)))
   (+ (- (log z) t) (* a (log t)))
   (- (+ (log y) (log (/ z (sqrt t)))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -60000000.0) || !(a <= 9.5e-38)) {
		tmp = (log(z) - t) + (a * log(t));
	} else {
		tmp = (log(y) + log((z / sqrt(t)))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-60000000.0d0)) .or. (.not. (a <= 9.5d-38))) then
        tmp = (log(z) - t) + (a * log(t))
    else
        tmp = (log(y) + log((z / sqrt(t)))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -60000000.0) || !(a <= 9.5e-38)) {
		tmp = (Math.log(z) - t) + (a * Math.log(t));
	} else {
		tmp = (Math.log(y) + Math.log((z / Math.sqrt(t)))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -60000000.0) or not (a <= 9.5e-38):
		tmp = (math.log(z) - t) + (a * math.log(t))
	else:
		tmp = (math.log(y) + math.log((z / math.sqrt(t)))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -60000000.0) || !(a <= 9.5e-38))
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	else
		tmp = Float64(Float64(log(y) + log(Float64(z / sqrt(t)))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -60000000.0) || ~((a <= 9.5e-38)))
		tmp = (log(z) - t) + (a * log(t));
	else
		tmp = (log(y) + log((z / sqrt(t)))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -60000000.0], N[Not[LessEqual[a, 9.5e-38]], $MachinePrecision]], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] + N[Log[N[(z / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6e7 or 9.5000000000000009e-38 < a

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 98.7%

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

      1. *-commutative 98.7%

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

   6. Simplified 98.7%

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

   if -6e7 < a < 9.5000000000000009e-38

   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. Step-by-step derivation

      1. associate--l+ 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+l+ 99.5%

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

      4. +-commutative 99.5%

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

      5. fma-def 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a around 0 99.3%

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

      1. associate-+r+ 99.3%

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

      2. +-commutative 99.3%

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

      3. +-commutative 99.3%

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

      4. log-prod 75.3%

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

      5. *-commutative 75.3%

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

      6. +-commutative 75.3%

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

   6. Simplified 75.3%

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

   7. Taylor expanded in z around 0 99.3%

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

      1. associate-+r+ 99.3%

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

      2. log-prod 75.3%

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

      3. log-pow 75.3%

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

      4. log-prod 70.5%

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

      5. associate-*l* 73.2%

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

   9. Simplified 73.2%

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

   10. Taylor expanded in x around 0 45.2%

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

       1. associate-*l* 49.3%

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

       2. log-prod 63.9%

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

       3. sqrt-div 63.9%

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

       4. metadata-eval 63.9%

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

       5. un-div-inv 63.9%

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

   12. Applied egg-rr 63.9%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -60000000 \lor \neg \left(a \leq 9.5 \cdot 10^{-38}\right):\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log \left(\frac{z}{\sqrt{t}}\right)\right) - t\\ \end{array} \]

Alternative 8: 69.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate--l+ 99.6%

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

   2. sub-neg 99.6%

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

   3. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 70.0%

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

   1. associate--l+ 70.0%

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

6. Simplified 70.0%

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

7. Final simplification 70.0%

   \[\leadsto \left(a + -0.5\right) \cdot \log t + \left(\log z + \left(\log y - t\right)\right) \]

Alternative 9: 87.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.0074:\\ \;\;\;\;\log \left(z \cdot \left(x + y\right)\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 0.0074)
   (+ (log (* z (+ x y))) (* (log t) (- a 0.5)))
   (+ (- (log z) t) (* a (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.0074) {
		tmp = log((z * (x + y))) + (log(t) * (a - 0.5));
	} else {
		tmp = (log(z) - t) + (a * log(t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 0.0074d0) then
        tmp = log((z * (x + y))) + (log(t) * (a - 0.5d0))
    else
        tmp = (log(z) - t) + (a * log(t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.0074) {
		tmp = Math.log((z * (x + y))) + (Math.log(t) * (a - 0.5));
	} else {
		tmp = (Math.log(z) - t) + (a * Math.log(t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 0.0074:
		tmp = math.log((z * (x + y))) + (math.log(t) * (a - 0.5))
	else:
		tmp = (math.log(z) - t) + (a * math.log(t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 0.0074)
		tmp = Float64(log(Float64(z * Float64(x + y))) + Float64(log(t) * Float64(a - 0.5)));
	else
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 0.0074)
		tmp = log((z * (x + y))) + (log(t) * (a - 0.5));
	else
		tmp = (log(z) - t) + (a * log(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 0.0074], N[(N[Log[N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 0.0074000000000000003

   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. Step-by-step derivation

      1. associate-+l- 99.3%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. associate--r+ 99.3%

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

      6. sub-neg 99.3%

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

      7. associate-+l- 99.3%

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

      8. neg-sub0 99.3%

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

      9. associate--r+ 99.3%

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

      10. unsub-neg 99.3%

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

      11. associate-+l- 99.3%

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

      12. neg-sub0 99.3%

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

      13. *-commutative 99.3%

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

      14. distribute-rgt-neg-in 99.3%

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

      15. fma-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in t around 0 98.1%

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

      1. +-commutative 98.1%

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

      2. log-prod 78.0%

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

      3. *-commutative 78.0%

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

      4. +-commutative 78.0%

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

      5. *-commutative 78.0%

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

   6. Simplified 78.0%

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

   if 0.0074000000000000003 < 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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around inf 97.0%

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

      1. *-commutative 97.0%

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

   6. Simplified 97.0%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.0074:\\ \;\;\;\;\log \left(z \cdot \left(x + y\right)\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 10: 66.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -45000000000 \lor \neg \left(a \leq 1.9 \cdot 10^{+22}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \log \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -45000000000.0) (not (<= a 1.9e+22)))
   (* a (log t))
   (+ (- (log z) t) (log (+ x y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -45000000000.0) || !(a <= 1.9e+22)) {
		tmp = a * log(t);
	} else {
		tmp = (log(z) - t) + log((x + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-45000000000.0d0)) .or. (.not. (a <= 1.9d+22))) then
        tmp = a * log(t)
    else
        tmp = (log(z) - t) + log((x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -45000000000.0) || !(a <= 1.9e+22)) {
		tmp = a * Math.log(t);
	} else {
		tmp = (Math.log(z) - t) + Math.log((x + y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -45000000000.0) or not (a <= 1.9e+22):
		tmp = a * math.log(t)
	else:
		tmp = (math.log(z) - t) + math.log((x + y))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -45000000000.0) || !(a <= 1.9e+22))
		tmp = Float64(a * log(t));
	else
		tmp = Float64(Float64(log(z) - t) + log(Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -45000000000.0) || ~((a <= 1.9e+22)))
		tmp = a * log(t);
	else
		tmp = (log(z) - t) + log((x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -45000000000.0], N[Not[LessEqual[a, 1.9e+22]], $MachinePrecision]], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.5e10 or 1.9000000000000002e22 < a

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. fma-def 99.7%

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

      3. sub-neg 99.7%

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

      4. metadata-eval 99.7%

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

      5. associate--l+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 75.1%

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

   5. Taylor expanded in a around inf 77.8%

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

      1. *-commutative 77.8%

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

   7. Simplified 77.8%

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

   if -4.5e10 < a < 1.9000000000000002e22

   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. Step-by-step derivation

      1. associate-+l- 99.5%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. associate--r+ 99.5%

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

      6. sub-neg 99.5%

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

      7. associate-+l- 99.5%

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

      8. neg-sub0 99.5%

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

      9. associate--r+ 99.5%

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

      10. unsub-neg 99.5%

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

      11. associate-+l- 99.5%

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

      12. neg-sub0 99.5%

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

      13. *-commutative 99.5%

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

      14. distribute-rgt-neg-in 99.5%

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

      15. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in t around inf 63.7%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -45000000000 \lor \neg \left(a \leq 1.9 \cdot 10^{+22}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \log \left(x + y\right)\\ \end{array} \]

Alternative 11: 71.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -60000000 \lor \neg \left(a \leq 5.4 \cdot 10^{-44}\right):\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{z \cdot y}{\sqrt{t}}\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -60000000.0) (not (<= a 5.4e-44)))
   (+ (- (log z) t) (* a (log t)))
   (- (log (/ (* z y) (sqrt t))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -60000000.0) || !(a <= 5.4e-44)) {
		tmp = (log(z) - t) + (a * log(t));
	} else {
		tmp = log(((z * y) / sqrt(t))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-60000000.0d0)) .or. (.not. (a <= 5.4d-44))) then
        tmp = (log(z) - t) + (a * log(t))
    else
        tmp = log(((z * y) / sqrt(t))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -60000000.0) || !(a <= 5.4e-44)) {
		tmp = (Math.log(z) - t) + (a * Math.log(t));
	} else {
		tmp = Math.log(((z * y) / Math.sqrt(t))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -60000000.0) or not (a <= 5.4e-44):
		tmp = (math.log(z) - t) + (a * math.log(t))
	else:
		tmp = math.log(((z * y) / math.sqrt(t))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -60000000.0) || !(a <= 5.4e-44))
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	else
		tmp = Float64(log(Float64(Float64(z * y) / sqrt(t))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -60000000.0) || ~((a <= 5.4e-44)))
		tmp = (log(z) - t) + (a * log(t));
	else
		tmp = log(((z * y) / sqrt(t))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -60000000.0], N[Not[LessEqual[a, 5.4e-44]], $MachinePrecision]], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(N[(z * y), $MachinePrecision] / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6e7 or 5.3999999999999998e-44 < a

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 97.5%

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

      1. *-commutative 97.5%

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

   6. Simplified 97.5%

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

   if -6e7 < a < 5.3999999999999998e-44

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-+l+ 99.5%

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

      4. +-commutative 99.5%

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

      5. fma-def 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a around 0 99.3%

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

      1. associate-+r+ 99.3%

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

      2. +-commutative 99.3%

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

      3. +-commutative 99.3%

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

      4. log-prod 76.4%

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

      5. *-commutative 76.4%

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

      6. +-commutative 76.4%

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

   6. Simplified 76.4%

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

   7. Taylor expanded in z around 0 99.3%

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

      1. associate-+r+ 99.3%

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

      2. log-prod 76.4%

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

      3. log-pow 76.4%

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

      4. log-prod 71.5%

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

      5. associate-*l* 74.2%

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

   9. Simplified 74.2%

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

   10. Taylor expanded in x around 0 46.2%

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

       1. sqrt-div 46.2%

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

       2. metadata-eval 46.2%

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

       3. un-div-inv 46.2%

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

   12. Applied egg-rr 46.2%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -60000000 \lor \neg \left(a \leq 5.4 \cdot 10^{-44}\right):\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{z \cdot y}{\sqrt{t}}\right) - t\\ \end{array} \]

Alternative 12: 73.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.005:\\ \;\;\;\;\log \left(z \cdot y\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 0.005)
   (+ (log (* z y)) (* (log t) (- a 0.5)))
   (+ (- (log z) t) (* a (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.005) {
		tmp = log((z * y)) + (log(t) * (a - 0.5));
	} else {
		tmp = (log(z) - t) + (a * log(t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 0.005d0) then
        tmp = log((z * y)) + (log(t) * (a - 0.5d0))
    else
        tmp = (log(z) - t) + (a * log(t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.005) {
		tmp = Math.log((z * y)) + (Math.log(t) * (a - 0.5));
	} else {
		tmp = (Math.log(z) - t) + (a * Math.log(t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 0.005:
		tmp = math.log((z * y)) + (math.log(t) * (a - 0.5))
	else:
		tmp = (math.log(z) - t) + (a * math.log(t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 0.005)
		tmp = Float64(log(Float64(z * y)) + Float64(log(t) * Float64(a - 0.5)));
	else
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 0.005)
		tmp = log((z * y)) + (log(t) * (a - 0.5));
	else
		tmp = (log(z) - t) + (a * log(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 0.005], N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 0.0050000000000000001

   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. Step-by-step derivation

      1. +-commutative 99.3%

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

      2. fma-def 99.3%

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

      3. sub-neg 99.3%

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

      4. metadata-eval 99.3%

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

      5. associate--l+ 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 68.2%

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

   5. Taylor expanded in z around inf 68.2%

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

      1. mul-1-neg 68.2%

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

      2. log-rec 68.2%

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

      3. remove-double-neg 68.2%

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

      4. log-prod 51.5%

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

   7. Simplified 51.5%

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

   8. Taylor expanded in t around 0 50.4%

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

   if 0.0050000000000000001 < 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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around inf 97.0%

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

      1. *-commutative 97.0%

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

   6. Simplified 97.0%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.005:\\ \;\;\;\;\log \left(z \cdot y\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 13: 76.2% 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. Step-by-step derivation

   1. associate--l+ 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-+l+ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-def 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a around inf 80.8%

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

   1. *-commutative 80.8%

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

6. Simplified 80.8%

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

7. Final simplification 80.8%

   \[\leadsto \left(\log z - t\right) + a \cdot \log t \]

Alternative 14: 62.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -43000000000 \lor \neg \left(a \leq 8 \cdot 10^{+21}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -43000000000.0) (not (<= a 8e+21))) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -43000000000.0) || !(a <= 8e+21)) {
		tmp = a * log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-43000000000.0d0)) .or. (.not. (a <= 8d+21))) then
        tmp = a * log(t)
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -43000000000.0) || !(a <= 8e+21)) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -43000000000.0) or not (a <= 8e+21):
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -43000000000.0) || !(a <= 8e+21))
		tmp = Float64(a * log(t));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -43000000000.0) || ~((a <= 8e+21)))
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -43000000000.0], N[Not[LessEqual[a, 8e+21]], $MachinePrecision]], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if a < -4.3e10 or 8e21 < a

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. fma-def 99.7%

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

      3. sub-neg 99.7%

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

      4. metadata-eval 99.7%

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

      5. associate--l+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 75.1%

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

   5. Taylor expanded in a around inf 77.8%

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

      1. *-commutative 77.8%

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

   7. Simplified 77.8%

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

   if -4.3e10 < a < 8e21

   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. Step-by-step derivation

      1. associate--l+ 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. fma-def 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in t around inf 56.5%

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

      1. neg-mul-1 56.5%

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

   6. Simplified 56.5%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -43000000000 \lor \neg \left(a \leq 8 \cdot 10^{+21}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 15: 37.7% accurate, 156.5× 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. Step-by-step derivation

   1. associate--l+ 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-+l+ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-def 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in t around inf 39.4%

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

   1. neg-mul-1 39.4%

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

6. Simplified 39.4%

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

7. Final simplification 39.4%

   \[\leadsto -t \]

Developer target: 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 2023224 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (+ (log (+ x y)) (+ (- (log z) t) (* (- a 0.5) (log t))))

  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))