Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 19.4s

Alternatives: 18

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-+l+ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-def 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. associate-+l+ 99.6%

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

   3. +-commutative 99.6%

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

   4. associate-+r- 99.6%

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

   5. fma-def 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. cancel-sign-sub 99.6%

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

   2. cancel-sign-sub-inv 99.6%

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

   3. associate--l+ 99.6%

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

   4. remove-double-neg 99.6%

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

   5. sub-neg 99.6%

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

   6. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 4: 64.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.085 \lor \neg \left(a \leq 0.72\right):\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log z + \left(\log \left(y \cdot {t}^{-0.5}\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -0.085) (not (<= a 0.72)))
   (- (+ (log y) (* a (log t))) t)
   (+ (log z) (- (log (* y (pow t -0.5))) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.085) || !(a <= 0.72)) {
		tmp = (log(y) + (a * log(t))) - t;
	} else {
		tmp = log(z) + (log((y * pow(t, -0.5))) - t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-0.085d0)) .or. (.not. (a <= 0.72d0))) then
        tmp = (log(y) + (a * log(t))) - t
    else
        tmp = log(z) + (log((y * (t ** (-0.5d0)))) - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.085) || !(a <= 0.72)) {
		tmp = (Math.log(y) + (a * Math.log(t))) - t;
	} else {
		tmp = Math.log(z) + (Math.log((y * Math.pow(t, -0.5))) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -0.085) or not (a <= 0.72):
		tmp = (math.log(y) + (a * math.log(t))) - t
	else:
		tmp = math.log(z) + (math.log((y * math.pow(t, -0.5))) - t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -0.085) || !(a <= 0.72))
		tmp = Float64(Float64(log(y) + Float64(a * log(t))) - t);
	else
		tmp = Float64(log(z) + Float64(log(Float64(y * (t ^ -0.5))) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -0.085) || ~((a <= 0.72)))
		tmp = (log(y) + (a * log(t))) - t;
	else
		tmp = log(z) + (log((y * (t ^ -0.5))) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -0.085], N[Not[LessEqual[a, 0.72]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[z], $MachinePrecision] + N[(N[Log[N[(y * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0850000000000000061 or 0.71999999999999997 < a

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r- 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 99.2%

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

      1. *-commutative 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in x around 0 74.3%

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

   if -0.0850000000000000061 < a < 0.71999999999999997

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. fma-def 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 73.7%

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

   5. Taylor expanded in a around 0 73.3%

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

      1. *-commutative 73.3%

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

   7. Simplified 73.3%

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

      1. associate-+l- 73.3%

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

      2. add-log-exp 73.3%

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

      3. pow-to-exp 73.3%

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

      4. sum-log 61.6%

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

   9. Applied egg-rr 61.6%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.085 \lor \neg \left(a \leq 0.72\right):\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log z + \left(\log \left(y \cdot {t}^{-0.5}\right) - t\right)\\ \end{array} \]

Alternative 5: 68.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 580:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 580.0)
   (+ (log y) (+ (log z) (* (log t) (- a 0.5))))
   (- (+ (log y) (* a (log t))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 580.0) {
		tmp = log(y) + (log(z) + (log(t) * (a - 0.5)));
	} else {
		tmp = (log(y) + (a * log(t))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 580.0d0) then
        tmp = log(y) + (log(z) + (log(t) * (a - 0.5d0)))
    else
        tmp = (log(y) + (a * log(t))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 580.0) {
		tmp = Math.log(y) + (Math.log(z) + (Math.log(t) * (a - 0.5)));
	} else {
		tmp = (Math.log(y) + (a * Math.log(t))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 580.0:
		tmp = math.log(y) + (math.log(z) + (math.log(t) * (a - 0.5)))
	else:
		tmp = (math.log(y) + (a * math.log(t))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 580.0)
		tmp = Float64(log(y) + Float64(log(z) + Float64(log(t) * Float64(a - 0.5))));
	else
		tmp = Float64(Float64(log(y) + Float64(a * log(t))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 580.0)
		tmp = log(y) + (log(z) + (log(t) * (a - 0.5)));
	else
		tmp = (log(y) + (a * log(t))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 580.0], 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[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 580

   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%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.3%

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

      4. +-commutative 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 68.5%

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

   5. Taylor expanded in t around 0 67.8%

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

   if 580 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r- 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around inf 99.4%

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

      1. *-commutative 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in x around 0 79.1%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 580:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \end{array} \]

Alternative 6: 69.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-+l+ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-def 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 74.0%

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

5. Final simplification 74.0%

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

Alternative 7: 59.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -6.8 \cdot 10^{+28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-228}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-153}:\\ \;\;\;\;\log \left(z \cdot y\right) + -0.5 \cdot \log t\\ \mathbf{elif}\;a \leq 8 \cdot 10^{+91}:\\ \;\;\;\;\log y - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= a -6.8e+28)
     t_1
     (if (<= a 4.2e-228)
       (- (log (+ x y)) t)
       (if (<= a 1.05e-153)
         (+ (log (* z y)) (* -0.5 (log t)))
         (if (<= a 8e+91) (- (log y) t) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if (a <= -6.8e+28) {
		tmp = t_1;
	} else if (a <= 4.2e-228) {
		tmp = log((x + y)) - t;
	} else if (a <= 1.05e-153) {
		tmp = log((z * y)) + (-0.5 * log(t));
	} else if (a <= 8e+91) {
		tmp = log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if (a <= (-6.8d+28)) then
        tmp = t_1
    else if (a <= 4.2d-228) then
        tmp = log((x + y)) - t
    else if (a <= 1.05d-153) then
        tmp = log((z * y)) + ((-0.5d0) * log(t))
    else if (a <= 8d+91) then
        tmp = log(y) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if (a <= -6.8e+28) {
		tmp = t_1;
	} else if (a <= 4.2e-228) {
		tmp = Math.log((x + y)) - t;
	} else if (a <= 1.05e-153) {
		tmp = Math.log((z * y)) + (-0.5 * Math.log(t));
	} else if (a <= 8e+91) {
		tmp = Math.log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if a <= -6.8e+28:
		tmp = t_1
	elif a <= 4.2e-228:
		tmp = math.log((x + y)) - t
	elif a <= 1.05e-153:
		tmp = math.log((z * y)) + (-0.5 * math.log(t))
	elif a <= 8e+91:
		tmp = math.log(y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (a <= -6.8e+28)
		tmp = t_1;
	elseif (a <= 4.2e-228)
		tmp = Float64(log(Float64(x + y)) - t);
	elseif (a <= 1.05e-153)
		tmp = Float64(log(Float64(z * y)) + Float64(-0.5 * log(t)));
	elseif (a <= 8e+91)
		tmp = Float64(log(y) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if (a <= -6.8e+28)
		tmp = t_1;
	elseif (a <= 4.2e-228)
		tmp = log((x + y)) - t;
	elseif (a <= 1.05e-153)
		tmp = log((z * y)) + (-0.5 * log(t));
	elseif (a <= 8e+91)
		tmp = log(y) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6.8e+28], t$95$1, If[LessEqual[a, 4.2e-228], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 1.05e-153], N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 8e+91], N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -6.8e28 or 8.00000000000000064e91 < a

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. fma-def 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 75.5%

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

   5. Taylor expanded in a around inf 83.6%

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

      1. *-commutative 83.6%

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

   7. Simplified 83.6%

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

   if -6.8e28 < a < 4.19999999999999982e-228

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r- 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 65.8%

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

      1. *-commutative 65.8%

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

   6. Simplified 65.8%

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

   7. Taylor expanded in a around 0 64.4%

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

      1. +-commutative 64.4%

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

   9. Simplified 64.4%

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

   if 4.19999999999999982e-228 < a < 1.05000000000000002e-153

   1. Initial program 99.0%

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

      1. associate--l+ 99.0%

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

      2. +-commutative 99.0%

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

      3. associate-+l+ 99.3%

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

      4. +-commutative 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 81.6%

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

   5. Taylor expanded in a around 0 81.6%

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

      1. *-commutative 81.6%

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

   7. Simplified 81.6%

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

   8. Taylor expanded in t around 0 68.7%

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

      1. associate-+r+ 68.9%

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

      2. log-prod 61.5%

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

      3. *-commutative 61.5%

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

   10. Simplified 61.5%

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

   if 1.05000000000000002e-153 < a < 8.00000000000000064e91

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. associate-+l+ 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. fma-def 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a around inf 75.4%

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

      1. *-commutative 75.4%

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

   6. Simplified 75.4%

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

   7. Taylor expanded in x around 0 63.0%

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

   8. Taylor expanded in a around 0 54.9%

      \[\leadsto \color{blue}{\log y - t} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{+28}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-228}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-153}:\\ \;\;\;\;\log \left(z \cdot y\right) + -0.5 \cdot \log t\\ \mathbf{elif}\;a \leq 8 \cdot 10^{+91}:\\ \;\;\;\;\log y - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t\\ \end{array} \]

Alternative 8: 60.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -5.8 \cdot 10^{+32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{-43}:\\ \;\;\;\;\log \left(\frac{y}{\frac{\sqrt{t}}{z}}\right) - t\\ \mathbf{elif}\;a \leq 4 \cdot 10^{+91}:\\ \;\;\;\;\left(t_1 + \log x\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= a -5.8e+32)
     t_1
     (if (<= a 2.8e-43)
       (- (log (/ y (/ (sqrt t) z))) t)
       (if (<= a 4e+91) (- (+ t_1 (log x)) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if (a <= -5.8e+32) {
		tmp = t_1;
	} else if (a <= 2.8e-43) {
		tmp = log((y / (sqrt(t) / z))) - t;
	} else if (a <= 4e+91) {
		tmp = (t_1 + log(x)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if (a <= (-5.8d+32)) then
        tmp = t_1
    else if (a <= 2.8d-43) then
        tmp = log((y / (sqrt(t) / z))) - t
    else if (a <= 4d+91) then
        tmp = (t_1 + log(x)) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if (a <= -5.8e+32) {
		tmp = t_1;
	} else if (a <= 2.8e-43) {
		tmp = Math.log((y / (Math.sqrt(t) / z))) - t;
	} else if (a <= 4e+91) {
		tmp = (t_1 + Math.log(x)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if a <= -5.8e+32:
		tmp = t_1
	elif a <= 2.8e-43:
		tmp = math.log((y / (math.sqrt(t) / z))) - t
	elif a <= 4e+91:
		tmp = (t_1 + math.log(x)) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (a <= -5.8e+32)
		tmp = t_1;
	elseif (a <= 2.8e-43)
		tmp = Float64(log(Float64(y / Float64(sqrt(t) / z))) - t);
	elseif (a <= 4e+91)
		tmp = Float64(Float64(t_1 + log(x)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if (a <= -5.8e+32)
		tmp = t_1;
	elseif (a <= 2.8e-43)
		tmp = log((y / (sqrt(t) / z))) - t;
	elseif (a <= 4e+91)
		tmp = (t_1 + log(x)) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -5.8e+32], t$95$1, If[LessEqual[a, 2.8e-43], N[(N[Log[N[(y / N[(N[Sqrt[t], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 4e+91], N[(N[(t$95$1 + N[Log[x], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.80000000000000006e32 or 4.00000000000000032e91 < a

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. fma-def 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 75.5%

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

   5. Taylor expanded in a around inf 83.6%

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

      1. *-commutative 83.6%

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

   7. Simplified 83.6%

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

   if -5.80000000000000006e32 < a < 2.7999999999999998e-43

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. associate-+l+ 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. fma-def 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a around 0 97.9%

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

      1. associate-+r+ 97.8%

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

      2. +-commutative 97.8%

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

      3. log-prod 72.6%

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

      4. *-commutative 72.6%

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

      5. +-commutative 72.6%

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

   6. Simplified 72.6%

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

   7. Taylor expanded in z around 0 97.9%

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

      1. associate-+r+ 97.8%

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

      2. +-commutative 97.8%

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

      3. log-prod 72.6%

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

      4. log-pow 72.6%

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

      5. log-prod 66.5%

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

      6. associate-*l* 66.6%

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

   9. Simplified 66.6%

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

   10. Taylor expanded in x around 0 47.9%

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

       1. *-un-lft-identity 47.9%

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

       2. log-prod 47.9%

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

       3. metadata-eval 47.9%

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

       4. *-commutative 47.9%

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

       5. sqrt-div 47.9%

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

       6. metadata-eval 47.9%

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

       7. un-div-inv 47.9%

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

   12. Applied egg-rr 47.9%

       \[\leadsto \color{blue}{\left(0 + \log \left(\frac{y \cdot z}{\sqrt{t}}\right)\right)} - t \]
   13. Step-by-step derivation

       1. +-lft-identity 47.9%

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

       2. associate-/l* 49.9%

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

   14. Simplified 49.9%

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

   if 2.7999999999999998e-43 < a < 4.00000000000000032e91

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. associate-+l+ 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. fma-def 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a around inf 86.6%

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

      1. *-commutative 86.6%

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

   6. Simplified 86.6%

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

   7. Taylor expanded in y around 0 81.9%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{+32}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{-43}:\\ \;\;\;\;\log \left(\frac{y}{\frac{\sqrt{t}}{z}}\right) - t\\ \mathbf{elif}\;a \leq 4 \cdot 10^{+91}:\\ \;\;\;\;\left(a \cdot \log t + \log x\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t\\ \end{array} \]

Alternative 9: 74.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.62 \cdot 10^{-12}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \log \left(z \cdot \left(x + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.62e-12)
   (+ (* (+ a -0.5) (log t)) (log (* z (+ x y))))
   (- (+ (log y) (* a (log t))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.62e-12) {
		tmp = ((a + -0.5) * log(t)) + log((z * (x + y)));
	} else {
		tmp = (log(y) + (a * log(t))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 1.62d-12) then
        tmp = ((a + (-0.5d0)) * log(t)) + log((z * (x + y)))
    else
        tmp = (log(y) + (a * log(t))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.62e-12) {
		tmp = ((a + -0.5) * Math.log(t)) + Math.log((z * (x + y)));
	} else {
		tmp = (Math.log(y) + (a * Math.log(t))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.62e-12:
		tmp = ((a + -0.5) * math.log(t)) + math.log((z * (x + y)))
	else:
		tmp = (math.log(y) + (a * math.log(t))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.62e-12)
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) + log(Float64(z * Float64(x + y))));
	else
		tmp = Float64(Float64(log(y) + Float64(a * log(t))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 1.62e-12)
		tmp = ((a + -0.5) * log(t)) + log((z * (x + y)));
	else
		tmp = (log(y) + (a * log(t))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.62e-12], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.62e-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. cancel-sign-sub 99.2%

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

      2. cancel-sign-sub-inv 99.2%

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

      3. associate--l+ 99.2%

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

      4. remove-double-neg 99.2%

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

      5. sub-neg 99.2%

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

      6. 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. Taylor expanded in t around 0 99.1%

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

      1. log-prod 75.7%

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

      2. +-commutative 75.7%

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

   6. Simplified 75.7%

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

   if 1.62e-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. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r- 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around inf 98.2%

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

      1. *-commutative 98.2%

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

   6. Simplified 98.2%

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

   7. Taylor expanded in x around 0 78.8%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.62 \cdot 10^{-12}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \log \left(z \cdot \left(x + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \end{array} \]

Alternative 10: 61.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{+17}:\\ \;\;\;\;\left(\log t \cdot \left(a - 0.5\right) + \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 9.5e+17)
   (- (+ (* (log t) (- a 0.5)) (log (* z y))) t)
   (- (+ (log y) (* a (log t))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 9.5e+17) {
		tmp = ((log(t) * (a - 0.5)) + log((z * y))) - t;
	} else {
		tmp = (log(y) + (a * log(t))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 9.5d+17) then
        tmp = ((log(t) * (a - 0.5d0)) + log((z * y))) - t
    else
        tmp = (log(y) + (a * log(t))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 9.5e+17) {
		tmp = ((Math.log(t) * (a - 0.5)) + Math.log((z * y))) - t;
	} else {
		tmp = (Math.log(y) + (a * Math.log(t))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 9.5e+17:
		tmp = ((math.log(t) * (a - 0.5)) + math.log((z * y))) - t
	else:
		tmp = (math.log(y) + (a * math.log(t))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 9.5e+17)
		tmp = Float64(Float64(Float64(log(t) * Float64(a - 0.5)) + log(Float64(z * y))) - t);
	else
		tmp = Float64(Float64(log(y) + Float64(a * log(t))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 9.5e+17)
		tmp = ((log(t) * (a - 0.5)) + log((z * y))) - t;
	else
		tmp = (log(y) + (a * log(t))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 9.5e+17], N[(N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.5e17

   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+ 99.3%

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

      2. associate-+l+ 99.3%

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

      3. +-commutative 99.3%

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

      4. associate-+r- 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. add-cbrt-cube 64.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\log \left(x + y\right) + \left(\mathsf{fma}\left(a + -0.5, \log t, \log z\right) - t\right)\right) \cdot \left(\log \left(x + y\right) + \left(\mathsf{fma}\left(a + -0.5, \log t, \log z\right) - t\right)\right)\right) \cdot \left(\log \left(x + y\right) + \left(\mathsf{fma}\left(a + -0.5, \log t, \log z\right) - t\right)\right)}} \]

      2. pow3 64.1%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\log \left(x + y\right) + \left(\mathsf{fma}\left(a + -0.5, \log t, \log z\right) - t\right)\right)}^{3}}} \]

      3. +-commutative 64.1%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(\mathsf{fma}\left(a + -0.5, \log t, \log z\right) - t\right) + \log \left(x + y\right)\right)}}^{3}} \]

      4. fma-udef 64.1%

         \[\leadsto \sqrt[3]{{\left(\left(\color{blue}{\left(\left(a + -0.5\right) \cdot \log t + \log z\right)} - t\right) + \log \left(x + y\right)\right)}^{3}} \]

      5. associate--l+ 64.1%

         \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(\left(a + -0.5\right) \cdot \log t + \left(\log z - t\right)\right)} + \log \left(x + y\right)\right)}^{3}} \]

      6. associate-+r+ 64.1%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(a + -0.5\right) \cdot \log t + \left(\left(\log z - t\right) + \log \left(x + y\right)\right)\right)}}^{3}} \]

      7. +-commutative 64.1%

         \[\leadsto \sqrt[3]{{\left(\left(a + -0.5\right) \cdot \log t + \color{blue}{\left(\log \left(x + y\right) + \left(\log z - t\right)\right)}\right)}^{3}} \]

      8. fma-def 64.1%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right) + \left(\log z - t\right)\right)\right)}}^{3}} \]

      9. associate-+r- 64.1%

         \[\leadsto \sqrt[3]{{\left(\mathsf{fma}\left(a + -0.5, \log t, \color{blue}{\left(\log \left(x + y\right) + \log z\right) - t}\right)\right)}^{3}} \]

      10. sum-log 45.1%

          \[\leadsto \sqrt[3]{{\left(\mathsf{fma}\left(a + -0.5, \log t, \color{blue}{\log \left(\left(x + y\right) \cdot z\right)} - t\right)\right)}^{3}} \]

   5. Applied egg-rr 45.1%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(a + -0.5, \log t, \log \left(\left(x + y\right) \cdot z\right) - t\right)\right)}^{3}}} \]

   6. Taylor expanded in x around 0 55.5%

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

   if 9.5e17 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r- 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around inf 99.9%

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

      1. *-commutative 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around 0 79.2%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{+17}:\\ \;\;\;\;\left(\log t \cdot \left(a - 0.5\right) + \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \end{array} \]

Alternative 11: 59.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.044 \lor \neg \left(a \leq 7 \cdot 10^{-44}\right):\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y}{\frac{\sqrt{t}}{z}}\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -0.044) (not (<= a 7e-44)))
   (- (+ (log y) (* a (log t))) t)
   (- (log (/ y (/ (sqrt t) z))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.044) || !(a <= 7e-44)) {
		tmp = (log(y) + (a * log(t))) - t;
	} else {
		tmp = log((y / (sqrt(t) / 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 <= (-0.044d0)) .or. (.not. (a <= 7d-44))) then
        tmp = (log(y) + (a * log(t))) - t
    else
        tmp = log((y / (sqrt(t) / 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 <= -0.044) || !(a <= 7e-44)) {
		tmp = (Math.log(y) + (a * Math.log(t))) - t;
	} else {
		tmp = Math.log((y / (Math.sqrt(t) / z))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -0.044) or not (a <= 7e-44):
		tmp = (math.log(y) + (a * math.log(t))) - t
	else:
		tmp = math.log((y / (math.sqrt(t) / z))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -0.044) || !(a <= 7e-44))
		tmp = Float64(Float64(log(y) + Float64(a * log(t))) - t);
	else
		tmp = Float64(log(Float64(y / Float64(sqrt(t) / z))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -0.044) || ~((a <= 7e-44)))
		tmp = (log(y) + (a * log(t))) - t;
	else
		tmp = log((y / (sqrt(t) / z))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -0.044], N[Not[LessEqual[a, 7e-44]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[N[(y / N[(N[Sqrt[t], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.043999999999999997 or 6.9999999999999995e-44 < a

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. associate-+l+ 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. fma-def 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a around inf 96.8%

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

      1. *-commutative 96.8%

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

   6. Simplified 96.8%

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

   7. Taylor expanded in x around 0 73.6%

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

   if -0.043999999999999997 < a < 6.9999999999999995e-44

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. associate-+l+ 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. fma-def 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a around 0 99.3%

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

      1. associate-+r+ 99.2%

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

      2. +-commutative 99.2%

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

      3. log-prod 73.9%

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

      4. *-commutative 73.9%

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

      5. +-commutative 73.9%

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

   6. Simplified 73.9%

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

   7. Taylor expanded in z around 0 99.3%

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

      1. associate-+r+ 99.2%

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

      2. +-commutative 99.2%

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

      3. log-prod 73.9%

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

      4. log-pow 73.9%

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

      5. log-prod 67.5%

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

      6. associate-*l* 67.7%

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

   9. Simplified 67.7%

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

   10. Taylor expanded in x around 0 49.8%

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

       1. *-un-lft-identity 49.8%

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

       2. log-prod 49.8%

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

       3. metadata-eval 49.8%

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

       4. *-commutative 49.8%

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

       5. sqrt-div 49.8%

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

       6. metadata-eval 49.8%

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

       7. un-div-inv 49.8%

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

   12. Applied egg-rr 49.8%

       \[\leadsto \color{blue}{\left(0 + \log \left(\frac{y \cdot z}{\sqrt{t}}\right)\right)} - t \]
   13. Step-by-step derivation

       1. +-lft-identity 49.8%

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

       2. associate-/l* 51.8%

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

   14. Simplified 51.8%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.044 \lor \neg \left(a \leq 7 \cdot 10^{-44}\right):\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y}{\frac{\sqrt{t}}{z}}\right) - t\\ \end{array} \]

Alternative 12: 57.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -9.2 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-43}:\\ \;\;\;\;\log \left(\frac{y}{\frac{\sqrt{t}}{z}}\right) - t\\ \mathbf{elif}\;a \leq 1.52 \cdot 10^{+93}:\\ \;\;\;\;\log y - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= a -9.2e+34)
     t_1
     (if (<= a 1.8e-43)
       (- (log (/ y (/ (sqrt t) z))) t)
       (if (<= a 1.52e+93) (- (log y) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if (a <= -9.2e+34) {
		tmp = t_1;
	} else if (a <= 1.8e-43) {
		tmp = log((y / (sqrt(t) / z))) - t;
	} else if (a <= 1.52e+93) {
		tmp = log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if (a <= (-9.2d+34)) then
        tmp = t_1
    else if (a <= 1.8d-43) then
        tmp = log((y / (sqrt(t) / z))) - t
    else if (a <= 1.52d+93) then
        tmp = log(y) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if (a <= -9.2e+34) {
		tmp = t_1;
	} else if (a <= 1.8e-43) {
		tmp = Math.log((y / (Math.sqrt(t) / z))) - t;
	} else if (a <= 1.52e+93) {
		tmp = Math.log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if a <= -9.2e+34:
		tmp = t_1
	elif a <= 1.8e-43:
		tmp = math.log((y / (math.sqrt(t) / z))) - t
	elif a <= 1.52e+93:
		tmp = math.log(y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (a <= -9.2e+34)
		tmp = t_1;
	elseif (a <= 1.8e-43)
		tmp = Float64(log(Float64(y / Float64(sqrt(t) / z))) - t);
	elseif (a <= 1.52e+93)
		tmp = Float64(log(y) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if (a <= -9.2e+34)
		tmp = t_1;
	elseif (a <= 1.8e-43)
		tmp = log((y / (sqrt(t) / z))) - t;
	elseif (a <= 1.52e+93)
		tmp = log(y) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -9.2e+34], t$95$1, If[LessEqual[a, 1.8e-43], N[(N[Log[N[(y / N[(N[Sqrt[t], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 1.52e+93], N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -9.1999999999999993e34 or 1.52e93 < a

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. fma-def 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 75.5%

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

   5. Taylor expanded in a around inf 83.6%

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

      1. *-commutative 83.6%

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

   7. Simplified 83.6%

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

   if -9.1999999999999993e34 < a < 1.7999999999999999e-43

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. associate-+l+ 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. fma-def 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a around 0 97.9%

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

      1. associate-+r+ 97.8%

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

      2. +-commutative 97.8%

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

      3. log-prod 72.6%

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

      4. *-commutative 72.6%

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

      5. +-commutative 72.6%

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

   6. Simplified 72.6%

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

   7. Taylor expanded in z around 0 97.9%

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

      1. associate-+r+ 97.8%

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

      2. +-commutative 97.8%

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

      3. log-prod 72.6%

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

      4. log-pow 72.6%

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

      5. log-prod 66.5%

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

      6. associate-*l* 66.6%

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

   9. Simplified 66.6%

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

   10. Taylor expanded in x around 0 47.9%

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

       1. *-un-lft-identity 47.9%

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

       2. log-prod 47.9%

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

       3. metadata-eval 47.9%

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

       4. *-commutative 47.9%

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

       5. sqrt-div 47.9%

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

       6. metadata-eval 47.9%

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

       7. un-div-inv 47.9%

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

   12. Applied egg-rr 47.9%

       \[\leadsto \color{blue}{\left(0 + \log \left(\frac{y \cdot z}{\sqrt{t}}\right)\right)} - t \]
   13. Step-by-step derivation

       1. +-lft-identity 47.9%

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

       2. associate-/l* 49.9%

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

   14. Simplified 49.9%

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

   if 1.7999999999999999e-43 < a < 1.52e93

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. associate-+l+ 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. fma-def 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a around inf 86.6%

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

      1. *-commutative 86.6%

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

   6. Simplified 86.6%

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

   7. Taylor expanded in x around 0 70.6%

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

   8. Taylor expanded in a around 0 55.8%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.2 \cdot 10^{+34}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-43}:\\ \;\;\;\;\log \left(\frac{y}{\frac{\sqrt{t}}{z}}\right) - t\\ \mathbf{elif}\;a \leq 1.52 \cdot 10^{+93}:\\ \;\;\;\;\log y - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t\\ \end{array} \]

Alternative 13: 61.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-12}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \log \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2.3e-12)
   (+ (* (+ a -0.5) (log t)) (log (* z y)))
   (- (+ (log y) (* a (log t))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.3e-12) {
		tmp = ((a + -0.5) * log(t)) + log((z * y));
	} else {
		tmp = (log(y) + (a * log(t))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 2.3d-12) then
        tmp = ((a + (-0.5d0)) * log(t)) + log((z * y))
    else
        tmp = (log(y) + (a * log(t))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.3e-12) {
		tmp = ((a + -0.5) * Math.log(t)) + Math.log((z * y));
	} else {
		tmp = (Math.log(y) + (a * Math.log(t))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 2.3e-12:
		tmp = ((a + -0.5) * math.log(t)) + math.log((z * y))
	else:
		tmp = (math.log(y) + (a * math.log(t))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.3e-12)
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) + log(Float64(z * y)));
	else
		tmp = Float64(Float64(log(y) + Float64(a * log(t))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 2.3e-12)
		tmp = ((a + -0.5) * log(t)) + log((z * y));
	else
		tmp = (log(y) + (a * log(t))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.3e-12], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.29999999999999989e-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%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.3%

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

      4. +-commutative 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 67.6%

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

   5. Taylor expanded in t around 0 67.5%

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

      1. associate-+r+ 67.5%

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

      2. log-prod 53.7%

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

      3. +-commutative 53.7%

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

      4. sub-neg 53.7%

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

      5. metadata-eval 53.7%

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

   7. Simplified 53.7%

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

   if 2.29999999999999989e-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. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r- 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around inf 98.2%

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

      1. *-commutative 98.2%

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

   6. Simplified 98.2%

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

   7. Taylor expanded in x around 0 78.8%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.3 \cdot 10^{-12}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \log \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \end{array} \]

Alternative 14: 64.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.14 \cdot 10^{+39} \lor \neg \left(a \leq 3.1 \cdot 10^{+91}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.14e+39) (not (<= a 3.1e+91)))
   (* a (log t))
   (- (log (+ x y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.14e+39) || !(a <= 3.1e+91)) {
		tmp = a * log(t);
	} 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 <= (-1.14d+39)) .or. (.not. (a <= 3.1d+91))) then
        tmp = a * log(t)
    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 <= -1.14e+39) || !(a <= 3.1e+91)) {
		tmp = a * Math.log(t);
	} else {
		tmp = Math.log((x + y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -1.14e+39) or not (a <= 3.1e+91):
		tmp = a * math.log(t)
	else:
		tmp = math.log((x + y)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.14e+39) || !(a <= 3.1e+91))
		tmp = Float64(a * log(t));
	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 <= -1.14e+39) || ~((a <= 3.1e+91)))
		tmp = a * log(t);
	else
		tmp = log((x + y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.14e+39], N[Not[LessEqual[a, 3.1e+91]], $MachinePrecision]], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.1399999999999999e39 or 3.09999999999999998e91 < a

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. fma-def 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 75.5%

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

   5. Taylor expanded in a around inf 83.6%

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

      1. *-commutative 83.6%

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

   7. Simplified 83.6%

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

   if -1.1399999999999999e39 < a < 3.09999999999999998e91

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. associate-+l+ 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. fma-def 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a around inf 64.8%

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

      1. *-commutative 64.8%

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

   6. Simplified 64.8%

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

   7. Taylor expanded in a around 0 60.3%

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

      1. +-commutative 60.3%

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

   9. Simplified 60.3%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.14 \cdot 10^{+39} \lor \neg \left(a \leq 3.1 \cdot 10^{+91}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \]

Alternative 15: 60.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.4 \cdot 10^{+27} \lor \neg \left(a \leq 2.95 \cdot 10^{+91}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -7.4e+27) (not (<= a 2.95e+91))) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -7.4e+27) || !(a <= 2.95e+91)) {
		tmp = a * log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-7.4d+27)) .or. (.not. (a <= 2.95d+91))) then
        tmp = a * log(t)
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -7.4e+27) || !(a <= 2.95e+91)) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -7.4e+27) or not (a <= 2.95e+91):
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -7.4e+27) || !(a <= 2.95e+91))
		tmp = Float64(a * log(t));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -7.4e+27) || ~((a <= 2.95e+91)))
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -7.4e+27], N[Not[LessEqual[a, 2.95e+91]], $MachinePrecision]], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if a < -7.40000000000000004e27 or 2.9500000000000001e91 < a

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. fma-def 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 75.5%

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

   5. Taylor expanded in a around inf 83.6%

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

      1. *-commutative 83.6%

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

   7. Simplified 83.6%

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

   if -7.40000000000000004e27 < a < 2.9500000000000001e91

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. associate-+l+ 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. fma-def 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in t around inf 54.9%

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

      1. neg-mul-1 54.9%

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

   6. Simplified 54.9%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.4 \cdot 10^{+27} \lor \neg \left(a \leq 2.95 \cdot 10^{+91}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 16: 55.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{+37} \lor \neg \left(a \leq 2.35 \cdot 10^{+92}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.35e+37) (not (<= a 2.35e+92))) (* a (log t)) (- (log y) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.35e+37) || !(a <= 2.35e+92)) {
		tmp = a * log(t);
	} 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 <= (-1.35d+37)) .or. (.not. (a <= 2.35d+92))) then
        tmp = a * log(t)
    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 <= -1.35e+37) || !(a <= 2.35e+92)) {
		tmp = a * Math.log(t);
	} else {
		tmp = Math.log(y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -1.35e+37) or not (a <= 2.35e+92):
		tmp = a * math.log(t)
	else:
		tmp = math.log(y) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.35e+37) || !(a <= 2.35e+92))
		tmp = Float64(a * log(t));
	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 <= -1.35e+37) || ~((a <= 2.35e+92)))
		tmp = a * log(t);
	else
		tmp = log(y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.35e+37], N[Not[LessEqual[a, 2.35e+92]], $MachinePrecision]], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.34999999999999993e37 or 2.35e92 < a

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. fma-def 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 75.5%

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

   5. Taylor expanded in a around inf 83.6%

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

      1. *-commutative 83.6%

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

   7. Simplified 83.6%

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

   if -1.34999999999999993e37 < a < 2.35e92

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. associate-+l+ 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.5%

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

      5. fma-def 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a around inf 64.8%

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

      1. *-commutative 64.8%

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

   6. Simplified 64.8%

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

   7. Taylor expanded in x around 0 52.8%

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

   8. Taylor expanded in a around 0 49.3%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{+37} \lor \neg \left(a \leq 2.35 \cdot 10^{+92}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log y - t\\ \end{array} \]

Alternative 17: 40.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 235:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (if (<= t 235.0) (log (+ x y)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 235.0) {
		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 <= 235.0d0) 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 <= 235.0) {
		tmp = Math.log((x + y));
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 235.0:
		tmp = math.log((x + y))
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 235.0)
		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 <= 235.0)
		tmp = log((x + y));
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 235.0], N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 235

   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%

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

      2. associate-+l+ 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-+r- 99.2%

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

      5. fma-def 99.2%

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

      6. sub-neg 99.2%

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

      7. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in a around inf 59.3%

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

      1. *-commutative 59.3%

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

   6. Simplified 59.3%

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

   7. Taylor expanded in a around 0 9.2%

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

      1. +-commutative 9.2%

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

   9. Simplified 9.2%

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

   10. Taylor expanded in t around 0 9.2%

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

       1. +-commutative 9.2%

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

   12. Simplified 9.2%

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

   if 235 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r- 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 75.6%

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

      1. neg-mul-1 75.6%

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

   6. Simplified 75.6%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 235:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 18: 36.8% accurate, 156.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. associate-+l+ 99.6%

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

   3. +-commutative 99.6%

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

   4. associate-+r- 99.6%

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

   5. fma-def 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in t around inf 39.1%

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

   1. neg-mul-1 39.1%

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

6. Simplified 39.1%

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

7. Final simplification 39.1%

   \[\leadsto -t \]

Developer target: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023293 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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