Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 16.9s

Alternatives: 17

Speedup: N/A×

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

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

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. *-commutative 99.6%

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

   6. distribute-rgt-neg-in 99.6%

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

   7. fma-undefine 99.6%

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

   8. sub-neg 99.6%

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

   9. +-commutative 99.6%

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

   10. distribute-neg-in 99.6%

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

   11. metadata-eval 99.6%

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

   12. metadata-eval 99.6%

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

   13. unsub-neg 99.6%

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

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. Add Preprocessing

5. Final simplification 99.6%

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

Alternative 2: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq -750 \lor \neg \left(t\_1 \leq 702\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(\log t \cdot \left(a + -0.5\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (or (<= t_1 -750.0) (not (<= t_1 702.0)))
     (- (+ (log y) (* (log t) a)) t)
     (+ (log (* (+ x y) z)) (- (* (log t) (+ a -0.5)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double tmp;
	if ((t_1 <= -750.0) || !(t_1 <= 702.0)) {
		tmp = (log(y) + (log(t) * a)) - t;
	} else {
		tmp = log(((x + y) * z)) + ((log(t) * (a + -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) :: t_1
    real(8) :: tmp
    t_1 = log((x + y)) + log(z)
    if ((t_1 <= (-750.0d0)) .or. (.not. (t_1 <= 702.0d0))) then
        tmp = (log(y) + (log(t) * a)) - t
    else
        tmp = log(((x + y) * z)) + ((log(t) * (a + (-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 t_1 = Math.log((x + y)) + Math.log(z);
	double tmp;
	if ((t_1 <= -750.0) || !(t_1 <= 702.0)) {
		tmp = (Math.log(y) + (Math.log(t) * a)) - t;
	} else {
		tmp = Math.log(((x + y) * z)) + ((Math.log(t) * (a + -0.5)) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log((x + y)) + math.log(z)
	tmp = 0
	if (t_1 <= -750.0) or not (t_1 <= 702.0):
		tmp = (math.log(y) + (math.log(t) * a)) - t
	else:
		tmp = math.log(((x + y) * z)) + ((math.log(t) * (a + -0.5)) - t)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	tmp = 0.0
	if ((t_1 <= -750.0) || !(t_1 <= 702.0))
		tmp = Float64(Float64(log(y) + Float64(log(t) * a)) - t);
	else
		tmp = Float64(log(Float64(Float64(x + y) * z)) + Float64(Float64(log(t) * Float64(a + -0.5)) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log((x + y)) + log(z);
	tmp = 0.0;
	if ((t_1 <= -750.0) || ~((t_1 <= 702.0)))
		tmp = (log(y) + (log(t) * a)) - t;
	else
		tmp = log(((x + y) * z)) + ((log(t) * (a + -0.5)) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -750.0], N[Not[LessEqual[t$95$1, 702.0]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 702 < (+.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. Step-by-step derivation

      1. remove-double-neg 99.8%

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

      2. associate--l+ 99.8%

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

      3. remove-double-neg 99.8%

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

      4. sub-neg 99.8%

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

      5. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 78.7%

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

   6. Taylor expanded in a around inf 66.7%

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

      1. *-commutative 66.7%

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

   8. Simplified 66.7%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 702

   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. remove-double-neg 99.5%

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

      2. associate--l+ 99.5%

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

      3. remove-double-neg 99.5%

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

      4. sub-neg 99.5%

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

      5. 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. Add Preprocessing
   5. Step-by-step derivation

      1. associate-+r- 99.5%

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

      2. associate-+l- 99.5%

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

      3. sum-log 99.7%

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

      4. *-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -750 \lor \neg \left(\log \left(x + y\right) + \log z \leq 702\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(\log t \cdot \left(a + -0.5\right) - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 425:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 425.0)
   (+ (log y) (+ (log z) (* (log t) (- a 0.5))))
   (- (+ (log y) (* (log t) a)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 425.0) {
		tmp = log(y) + (log(z) + (log(t) * (a - 0.5)));
	} else {
		tmp = (log(y) + (log(t) * a)) - 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 <= 425.0d0) then
        tmp = log(y) + (log(z) + (log(t) * (a - 0.5d0)))
    else
        tmp = (log(y) + (log(t) * a)) - 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 <= 425.0) {
		tmp = Math.log(y) + (Math.log(z) + (Math.log(t) * (a - 0.5)));
	} else {
		tmp = (Math.log(y) + (Math.log(t) * a)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 425.0:
		tmp = math.log(y) + (math.log(z) + (math.log(t) * (a - 0.5)))
	else:
		tmp = (math.log(y) + (math.log(t) * a)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 425.0)
		tmp = Float64(log(y) + Float64(log(z) + Float64(log(t) * Float64(a - 0.5))));
	else
		tmp = Float64(Float64(log(y) + Float64(log(t) * a)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 425.0)
		tmp = log(y) + (log(z) + (log(t) * (a - 0.5)));
	else
		tmp = (log(y) + (log(t) * a)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 425.0], N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 425

   1. Initial program 99.2%

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

      1. remove-double-neg 99.2%

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

      2. associate--l+ 99.2%

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

      3. remove-double-neg 99.2%

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

      4. sub-neg 99.2%

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

      5. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in x around 0 66.2%

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

   6. Taylor expanded in t around 0 65.8%

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

   if 425 < 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. remove-double-neg 99.9%

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

      2. associate--l+ 99.9%

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

      3. remove-double-neg 99.9%

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

      4. sub-neg 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 77.6%

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

   6. Taylor expanded in a around inf 77.6%

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

      1. *-commutative 77.6%

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

   8. Simplified 77.6%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 425:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)) + (log(t) * (a + (-0.5d0)))
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)) + (Math.log(t) * (a + -0.5));
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $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. remove-double-neg 99.6%

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

   2. associate--l+ 99.6%

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

   3. remove-double-neg 99.6%

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

   4. sub-neg 99.6%

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

   5. 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. Add Preprocessing

5. Final simplification 99.6%

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

Alternative 5: 68.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $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. 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. *-commutative 99.6%

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

   6. distribute-rgt-neg-in 99.6%

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

   7. fma-undefine 99.6%

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

   8. sub-neg 99.6%

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

   9. +-commutative 99.6%

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

   10. distribute-neg-in 99.6%

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

   11. metadata-eval 99.6%

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

   12. metadata-eval 99.6%

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

   13. unsub-neg 99.6%

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

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. Add Preprocessing

5. Taylor expanded in x around 0 71.8%

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

6. Final simplification 71.8%

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

Alternative 6: 68.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. remove-double-neg 99.6%

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

   2. associate--l+ 99.6%

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

   3. remove-double-neg 99.6%

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

   4. sub-neg 99.6%

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

   5. 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. Add Preprocessing

5. Taylor expanded in x around 0 71.9%

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

6. Final simplification 71.9%

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

Alternative 7: 58.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0265:\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{elif}\;a \leq 4.9 \cdot 10^{-5}:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{\left(a + -0.5\right)}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\log t + \frac{\log y}{a}\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -0.0265)
   (- (+ (log y) (* (log t) a)) t)
   (if (<= a 4.9e-5)
     (- (log (* y (* z (pow t (+ a -0.5))))) t)
     (- (* a (+ (log t) (/ (log y) a))) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -0.0265) {
		tmp = (log(y) + (log(t) * a)) - t;
	} else if (a <= 4.9e-5) {
		tmp = log((y * (z * pow(t, (a + -0.5))))) - t;
	} else {
		tmp = (a * (log(t) + (log(y) / a))) - 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 <= (-0.0265d0)) then
        tmp = (log(y) + (log(t) * a)) - t
    else if (a <= 4.9d-5) then
        tmp = log((y * (z * (t ** (a + (-0.5d0)))))) - t
    else
        tmp = (a * (log(t) + (log(y) / a))) - 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 <= -0.0265) {
		tmp = (Math.log(y) + (Math.log(t) * a)) - t;
	} else if (a <= 4.9e-5) {
		tmp = Math.log((y * (z * Math.pow(t, (a + -0.5))))) - t;
	} else {
		tmp = (a * (Math.log(t) + (Math.log(y) / a))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -0.0265:
		tmp = (math.log(y) + (math.log(t) * a)) - t
	elif a <= 4.9e-5:
		tmp = math.log((y * (z * math.pow(t, (a + -0.5))))) - t
	else:
		tmp = (a * (math.log(t) + (math.log(y) / a))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -0.0265)
		tmp = Float64(Float64(log(y) + Float64(log(t) * a)) - t);
	elseif (a <= 4.9e-5)
		tmp = Float64(log(Float64(y * Float64(z * (t ^ Float64(a + -0.5))))) - t);
	else
		tmp = Float64(Float64(a * Float64(log(t) + Float64(log(y) / a))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -0.0265)
		tmp = (log(y) + (log(t) * a)) - t;
	elseif (a <= 4.9e-5)
		tmp = log((y * (z * (t ^ (a + -0.5))))) - t;
	else
		tmp = (a * (log(t) + (log(y) / a))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -0.0265], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 4.9e-5], N[(N[Log[N[(y * N[(z * N[Power[t, N[(a + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], N[(N[(a * N[(N[Log[t], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.0264999999999999993

   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. remove-double-neg 99.7%

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

      2. associate--l+ 99.7%

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

      3. remove-double-neg 99.7%

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

      4. sub-neg 99.7%

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

      5. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 78.9%

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

   6. Taylor expanded in a around inf 78.4%

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

      1. *-commutative 78.4%

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

   8. Simplified 78.4%

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

   if -0.0264999999999999993 < a < 4.9e-5

   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. remove-double-neg 99.4%

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

      2. associate--l+ 99.4%

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

      3. remove-double-neg 99.4%

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

      4. sub-neg 99.4%

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

      5. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 64.4%

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

      1. add-log-exp 57.6%

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

      2. sum-log 42.8%

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

      3. exp-sum 42.8%

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

      4. add-exp-log 42.9%

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

      5. exp-to-pow 43.0%

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

      6. sub-neg 43.0%

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

      7. metadata-eval 43.0%

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

   7. Applied egg-rr 43.0%

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

   if 4.9e-5 < 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. remove-double-neg 99.7%

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

      2. associate--l+ 99.7%

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

      3. remove-double-neg 99.7%

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

      4. sub-neg 99.7%

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

      5. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 78.5%

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

   6. Taylor expanded in a around inf 77.1%

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

      1. *-commutative 77.1%

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

   8. Simplified 77.1%

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

   9. Taylor expanded in a around inf 77.2%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0265:\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{elif}\;a \leq 4.9 \cdot 10^{-5}:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{\left(a + -0.5\right)}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\log t + \frac{\log y}{a}\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{+25} \lor \neg \left(a \leq 1.8 \cdot 10^{+66}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -2.6e+25) (not (<= a 1.8e+66)))
   (* (log t) a)
   (+ (log (+ x y)) (- (log z) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2.6e+25) || !(a <= 1.8e+66)) {
		tmp = log(t) * a;
	} else {
		tmp = log((x + y)) + (log(z) - 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 <= (-2.6d+25)) .or. (.not. (a <= 1.8d+66))) then
        tmp = log(t) * a
    else
        tmp = log((x + y)) + (log(z) - 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 <= -2.6e+25) || !(a <= 1.8e+66)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = Math.log((x + y)) + (Math.log(z) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -2.6e+25) or not (a <= 1.8e+66):
		tmp = math.log(t) * a
	else:
		tmp = math.log((x + y)) + (math.log(z) - t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -2.6e+25) || !(a <= 1.8e+66))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(log(Float64(x + y)) + Float64(log(z) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -2.6e+25) || ~((a <= 1.8e+66)))
		tmp = log(t) * a;
	else
		tmp = log((x + y)) + (log(z) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -2.6e+25], N[Not[LessEqual[a, 1.8e+66]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.5999999999999998e25 or 1.8e66 < 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. remove-double-neg 99.7%

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

      2. associate--l+ 99.7%

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

      3. remove-double-neg 99.7%

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

      4. sub-neg 99.7%

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

      5. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 79.8%

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

   6. Taylor expanded in t around inf 55.5%

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

      1. sub-neg 55.5%

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

      2. associate-+r+ 55.5%

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

      3. mul-1-neg 55.5%

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

      4. log-rec 55.5%

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

      5. sub-neg 55.5%

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

      6. metadata-eval 55.5%

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

      7. associate-/l* 55.4%

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

      8. distribute-lft-neg-in 55.4%

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

      9. remove-double-neg 55.4%

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

      10. metadata-eval 55.4%

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

   8. Simplified 55.4%

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

   9. Taylor expanded in a around inf 84.8%

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

       1. *-commutative 84.8%

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

   11. Simplified 84.8%

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

   if -2.5999999999999998e25 < a < 1.8e66

   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. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-undefine 99.5%

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

      8. sub-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. distribute-neg-in 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. unsub-neg 99.5%

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

   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. Add Preprocessing

   5. Taylor expanded in t around inf 60.2%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.6 \cdot 10^{+25} \lor \neg \left(a \leq 1.8 \cdot 10^{+66}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4.9 \cdot 10^{-12}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 4.9e-12)
   (+ (log (* (+ x y) z)) (* (log t) (- a 0.5)))
   (- (+ (log y) (* (log t) a)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 4.9e-12) {
		tmp = log(((x + y) * z)) + (log(t) * (a - 0.5));
	} else {
		tmp = (log(y) + (log(t) * a)) - 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 <= 4.9d-12) then
        tmp = log(((x + y) * z)) + (log(t) * (a - 0.5d0))
    else
        tmp = (log(y) + (log(t) * a)) - 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 <= 4.9e-12) {
		tmp = Math.log(((x + y) * z)) + (Math.log(t) * (a - 0.5));
	} else {
		tmp = (Math.log(y) + (Math.log(t) * a)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 4.9e-12:
		tmp = math.log(((x + y) * z)) + (math.log(t) * (a - 0.5))
	else:
		tmp = (math.log(y) + (math.log(t) * a)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 4.9e-12)
		tmp = Float64(log(Float64(Float64(x + y) * z)) + Float64(log(t) * Float64(a - 0.5)));
	else
		tmp = Float64(Float64(log(y) + Float64(log(t) * a)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 4.9e-12)
		tmp = log(((x + y) * z)) + (log(t) * (a - 0.5));
	else
		tmp = (log(y) + (log(t) * a)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 4.9e-12], N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.89999999999999972e-12

   1. Initial program 99.2%

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

      1. associate-+l- 99.2%

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

      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. *-commutative 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. fma-undefine 99.3%

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

      8. sub-neg 99.3%

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

      9. +-commutative 99.3%

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

      10. distribute-neg-in 99.3%

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

      11. metadata-eval 99.3%

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

      12. metadata-eval 99.3%

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

      13. unsub-neg 99.3%

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

   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. Add Preprocessing

   5. Taylor expanded in t around inf 72.2%

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

      1. mul-1-neg 72.2%

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

      2. unsub-neg 72.2%

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

      3. associate-/l* 72.1%

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

      4. log-rec 72.1%

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

   7. Simplified 72.1%

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

      1. associate-+r- 72.1%

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

      2. sum-log 61.8%

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

      3. +-commutative 61.8%

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

      4. cancel-sign-sub 61.8%

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

   9. Applied egg-rr 61.8%

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

   10. Taylor expanded in t around 0 82.2%

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

   if 4.89999999999999972e-12 < 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. remove-double-neg 99.9%

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

      2. associate--l+ 99.9%

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

      3. remove-double-neg 99.9%

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

      4. sub-neg 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 77.3%

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

   6. Taylor expanded in a around inf 75.9%

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

      1. *-commutative 75.9%

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

   8. Simplified 75.9%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.9 \cdot 10^{-12}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+26} \lor \neg \left(a \leq 2.4 \cdot 10^{+65}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -3.8e+26) (not (<= a 2.4e+65)))
   (* (log t) a)
   (- (+ (log z) (log y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3.8e+26) || !(a <= 2.4e+65)) {
		tmp = log(t) * a;
	} else {
		tmp = (log(z) + log(y)) - 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 <= (-3.8d+26)) .or. (.not. (a <= 2.4d+65))) then
        tmp = log(t) * a
    else
        tmp = (log(z) + log(y)) - 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 <= -3.8e+26) || !(a <= 2.4e+65)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = (Math.log(z) + Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -3.8e+26) or not (a <= 2.4e+65):
		tmp = math.log(t) * a
	else:
		tmp = (math.log(z) + math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -3.8e+26) || !(a <= 2.4e+65))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(Float64(log(z) + log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -3.8e+26) || ~((a <= 2.4e+65)))
		tmp = log(t) * a;
	else
		tmp = (log(z) + log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -3.8e+26], N[Not[LessEqual[a, 2.4e+65]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.8000000000000002e26 or 2.4000000000000002e65 < 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. remove-double-neg 99.7%

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

      2. associate--l+ 99.7%

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

      3. remove-double-neg 99.7%

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

      4. sub-neg 99.7%

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

      5. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 79.8%

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

   6. Taylor expanded in t around inf 55.5%

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

      1. sub-neg 55.5%

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

      2. associate-+r+ 55.5%

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

      3. mul-1-neg 55.5%

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

      4. log-rec 55.5%

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

      5. sub-neg 55.5%

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

      6. metadata-eval 55.5%

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

      7. associate-/l* 55.4%

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

      8. distribute-lft-neg-in 55.4%

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

      9. remove-double-neg 55.4%

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

      10. metadata-eval 55.4%

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

   8. Simplified 55.4%

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

   9. Taylor expanded in a around inf 84.8%

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

       1. *-commutative 84.8%

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

   11. Simplified 84.8%

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

   if -3.8000000000000002e26 < a < 2.4000000000000002e65

   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. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-undefine 99.5%

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

      8. sub-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. distribute-neg-in 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. unsub-neg 99.5%

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

   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. Add Preprocessing

   5. Taylor expanded in t around inf 60.2%

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

   6. Taylor expanded in x around 0 46.0%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+26} \lor \neg \left(a \leq 2.4 \cdot 10^{+65}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. remove-double-neg 99.6%

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

   2. associate--l+ 99.6%

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

   3. remove-double-neg 99.6%

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

   4. sub-neg 99.6%

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

   5. 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. Add Preprocessing

5. Taylor expanded in x around 0 71.9%

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

6. Taylor expanded in a around inf 61.9%

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

   1. *-commutative 61.9%

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

8. Simplified 61.9%

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

9. Final simplification 61.9%

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

Alternative 12: 65.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.3 \cdot 10^{+23} \lor \neg \left(a \leq 1.25 \cdot 10^{+65}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{\log \left(x + y\right)}{t} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -5.3e+23) (not (<= a 1.25e+65)))
   (* (log t) a)
   (* t (+ (/ (log (+ x y)) t) -1.0))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -5.3e+23) || !(a <= 1.25e+65)) {
		tmp = log(t) * a;
	} else {
		tmp = t * ((log((x + y)) / t) + -1.0);
	}
	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 <= (-5.3d+23)) .or. (.not. (a <= 1.25d+65))) then
        tmp = log(t) * a
    else
        tmp = t * ((log((x + y)) / t) + (-1.0d0))
    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 <= -5.3e+23) || !(a <= 1.25e+65)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = t * ((Math.log((x + y)) / t) + -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -5.3e+23) or not (a <= 1.25e+65):
		tmp = math.log(t) * a
	else:
		tmp = t * ((math.log((x + y)) / t) + -1.0)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -5.3e+23) || !(a <= 1.25e+65))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(t * Float64(Float64(log(Float64(x + y)) / t) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -5.3e+23) || ~((a <= 1.25e+65)))
		tmp = log(t) * a;
	else
		tmp = t * ((log((x + y)) / t) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -5.3e+23], N[Not[LessEqual[a, 1.25e+65]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(t * N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.3000000000000001e23 or 1.24999999999999993e65 < 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. remove-double-neg 99.7%

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

      2. associate--l+ 99.7%

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

      3. remove-double-neg 99.7%

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

      4. sub-neg 99.7%

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

      5. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 79.8%

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

   6. Taylor expanded in t around inf 55.5%

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

      1. sub-neg 55.5%

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

      2. associate-+r+ 55.5%

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

      3. mul-1-neg 55.5%

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

      4. log-rec 55.5%

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

      5. sub-neg 55.5%

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

      6. metadata-eval 55.5%

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

      7. associate-/l* 55.4%

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

      8. distribute-lft-neg-in 55.4%

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

      9. remove-double-neg 55.4%

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

      10. metadata-eval 55.4%

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

   8. Simplified 55.4%

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

   9. Taylor expanded in a around inf 84.8%

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

       1. *-commutative 84.8%

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

   11. Simplified 84.8%

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

   if -5.3000000000000001e23 < a < 1.24999999999999993e65

   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. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-undefine 99.5%

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

      8. sub-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. distribute-neg-in 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. unsub-neg 99.5%

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

   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. Add Preprocessing

   5. Taylor expanded in t around inf 58.6%

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

      1. neg-mul-1 58.6%

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

   7. Simplified 58.6%

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

   8. Taylor expanded in t around inf 58.6%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.3 \cdot 10^{+23} \lor \neg \left(a \leq 1.25 \cdot 10^{+65}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{\log \left(x + y\right)}{t} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{+23} \lor \neg \left(a \leq 1.5 \cdot 10^{+65}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -5.2e+23) (not (<= a 1.5e+65)))
   (* (log t) a)
   (- (log (+ x y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -5.2e+23) || !(a <= 1.5e+65)) {
		tmp = log(t) * a;
	} else {
		tmp = log((x + y)) - 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 <= (-5.2d+23)) .or. (.not. (a <= 1.5d+65))) then
        tmp = log(t) * a
    else
        tmp = log((x + y)) - 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 <= -5.2e+23) || !(a <= 1.5e+65)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = Math.log((x + y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -5.2e+23) or not (a <= 1.5e+65):
		tmp = math.log(t) * a
	else:
		tmp = math.log((x + y)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -5.2e+23) || !(a <= 1.5e+65))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(log(Float64(x + y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -5.2e+23) || ~((a <= 1.5e+65)))
		tmp = log(t) * a;
	else
		tmp = log((x + y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -5.2e+23], N[Not[LessEqual[a, 1.5e+65]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.19999999999999983e23 or 1.5000000000000001e65 < 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. remove-double-neg 99.7%

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

      2. associate--l+ 99.7%

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

      3. remove-double-neg 99.7%

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

      4. sub-neg 99.7%

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

      5. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 79.8%

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

   6. Taylor expanded in t around inf 55.5%

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

      1. sub-neg 55.5%

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

      2. associate-+r+ 55.5%

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

      3. mul-1-neg 55.5%

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

      4. log-rec 55.5%

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

      5. sub-neg 55.5%

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

      6. metadata-eval 55.5%

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

      7. associate-/l* 55.4%

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

      8. distribute-lft-neg-in 55.4%

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

      9. remove-double-neg 55.4%

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

      10. metadata-eval 55.4%

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

   8. Simplified 55.4%

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

   9. Taylor expanded in a around inf 84.8%

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

       1. *-commutative 84.8%

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

   11. Simplified 84.8%

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

   if -5.19999999999999983e23 < a < 1.5000000000000001e65

   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. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-undefine 99.5%

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

      8. sub-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. distribute-neg-in 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. unsub-neg 99.5%

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

   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. Add Preprocessing

   5. Taylor expanded in t around inf 58.6%

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

      1. neg-mul-1 58.6%

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

   7. Simplified 58.6%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{+23} \lor \neg \left(a \leq 1.5 \cdot 10^{+65}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{+27} \lor \neg \left(a \leq 1.3 \cdot 10^{+65}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -7.8e+27) (not (<= a 1.3e+65))) (* (log t) a) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -7.8e+27) || !(a <= 1.3e+65)) {
		tmp = log(t) * a;
	} 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 <= (-7.8d+27)) .or. (.not. (a <= 1.3d+65))) then
        tmp = log(t) * a
    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 <= -7.8e+27) || !(a <= 1.3e+65)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -7.8e+27) or not (a <= 1.3e+65):
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -7.8e+27) || !(a <= 1.3e+65))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -7.8e+27) || ~((a <= 1.3e+65)))
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -7.8e+27], N[Not[LessEqual[a, 1.3e+65]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if a < -7.7999999999999997e27 or 1.30000000000000001e65 < 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. remove-double-neg 99.7%

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

      2. associate--l+ 99.7%

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

      3. remove-double-neg 99.7%

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

      4. sub-neg 99.7%

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

      5. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 79.8%

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

   6. Taylor expanded in t around inf 55.5%

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

      1. sub-neg 55.5%

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

      2. associate-+r+ 55.5%

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

      3. mul-1-neg 55.5%

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

      4. log-rec 55.5%

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

      5. sub-neg 55.5%

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

      6. metadata-eval 55.5%

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

      7. associate-/l* 55.4%

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

      8. distribute-lft-neg-in 55.4%

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

      9. remove-double-neg 55.4%

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

      10. metadata-eval 55.4%

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

   8. Simplified 55.4%

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

   9. Taylor expanded in a around inf 84.8%

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

       1. *-commutative 84.8%

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

   11. Simplified 84.8%

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

   if -7.7999999999999997e27 < a < 1.30000000000000001e65

   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. remove-double-neg 99.5%

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

      2. associate--l+ 99.5%

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

      3. remove-double-neg 99.5%

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

      4. sub-neg 99.5%

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

      5. 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. Add Preprocessing

   5. Taylor expanded in x around 0 65.9%

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

   6. Taylor expanded in t around inf 53.4%

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

      1. neg-mul-1 53.4%

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

   8. Simplified 53.4%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{+27} \lor \neg \left(a \leq 1.3 \cdot 10^{+65}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{+25} \lor \neg \left(a \leq 4.8 \cdot 10^{+65}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -5.5e+25) (not (<= a 4.8e+65))) (* (log t) a) (- (log y) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -5.5e+25) || !(a <= 4.8e+65)) {
		tmp = log(t) * a;
	} else {
		tmp = log(y) - 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 <= (-5.5d+25)) .or. (.not. (a <= 4.8d+65))) then
        tmp = log(t) * a
    else
        tmp = log(y) - 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 <= -5.5e+25) || !(a <= 4.8e+65)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = Math.log(y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -5.5e+25) or not (a <= 4.8e+65):
		tmp = math.log(t) * a
	else:
		tmp = math.log(y) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -5.5e+25) || !(a <= 4.8e+65))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(log(y) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -5.5e+25) || ~((a <= 4.8e+65)))
		tmp = log(t) * a;
	else
		tmp = log(y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -5.5e+25], N[Not[LessEqual[a, 4.8e+65]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.50000000000000018e25 or 4.8000000000000003e65 < 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. remove-double-neg 99.7%

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

      2. associate--l+ 99.7%

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

      3. remove-double-neg 99.7%

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

      4. sub-neg 99.7%

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

      5. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 79.8%

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

   6. Taylor expanded in t around inf 55.5%

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

      1. sub-neg 55.5%

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

      2. associate-+r+ 55.5%

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

      3. mul-1-neg 55.5%

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

      4. log-rec 55.5%

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

      5. sub-neg 55.5%

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

      6. metadata-eval 55.5%

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

      7. associate-/l* 55.4%

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

      8. distribute-lft-neg-in 55.4%

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

      9. remove-double-neg 55.4%

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

      10. metadata-eval 55.4%

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

   8. Simplified 55.4%

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

   9. Taylor expanded in a around inf 84.8%

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

       1. *-commutative 84.8%

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

   11. Simplified 84.8%

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

   if -5.50000000000000018e25 < a < 4.8000000000000003e65

   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. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-undefine 99.5%

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

      8. sub-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. distribute-neg-in 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. unsub-neg 99.5%

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

   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. Add Preprocessing

   5. Taylor expanded in t around inf 58.6%

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

      1. neg-mul-1 58.6%

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

   7. Simplified 58.6%

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

   8. Taylor expanded in x around 0 44.9%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{+25} \lor \neg \left(a \leq 4.8 \cdot 10^{+65}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 40.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 26500:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 26500.0) (log (+ x y)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 26500.0) {
		tmp = log((x + y));
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 26500.0d0) then
        tmp = log((x + y))
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 26500.0) {
		tmp = Math.log((x + y));
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 26500.0:
		tmp = math.log((x + y))
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 26500.0)
		tmp = log(Float64(x + y));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 26500.0)
		tmp = log((x + y));
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 26500.0], N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 26500

   1. Initial program 99.2%

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

      1. associate-+l- 99.2%

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

      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. *-commutative 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. fma-undefine 99.3%

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

      8. sub-neg 99.3%

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

      9. +-commutative 99.3%

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

      10. distribute-neg-in 99.3%

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

      11. metadata-eval 99.3%

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

      12. metadata-eval 99.3%

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

      13. unsub-neg 99.3%

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

   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. Add Preprocessing

   5. Taylor expanded in t around inf 9.1%

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

      1. neg-mul-1 9.1%

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

   7. Simplified 9.1%

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

   8. Taylor expanded in t around 0 9.1%

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

      1. +-commutative 9.1%

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

   10. Simplified 9.1%

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

   if 26500 < 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. remove-double-neg 99.9%

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

      2. associate--l+ 99.9%

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

      3. remove-double-neg 99.9%

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

      4. sub-neg 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 78.2%

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

   6. Taylor expanded in t around inf 73.2%

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

      1. neg-mul-1 73.2%

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

   8. Simplified 73.2%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 26500:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 37.5% 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. remove-double-neg 99.6%

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

   2. associate--l+ 99.6%

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

   3. remove-double-neg 99.6%

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

   4. sub-neg 99.6%

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

   5. 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. Add Preprocessing

5. Taylor expanded in x around 0 71.9%

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

6. Taylor expanded in t around inf 37.5%

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

   1. neg-mul-1 37.5%

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

8. Simplified 37.5%

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

9. Final simplification 37.5%

   \[\leadsto -t \]
10. Add Preprocessing

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 2024067 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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