Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 19.3s

Alternatives: 16

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

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

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)) + fma(Float64(a + -0.5), log(t), Float64(log(z) - t)))
end

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
2. 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. fma-def 99.6%

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

   5. remove-double-neg 99.6%

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

   6. remove-double-neg 99.6%

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

   7. sub-neg 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 66.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log z \leq 106:\\ \;\;\;\;\left(\log t \cdot \left(a - 0.5\right) + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (log z) 106.0)
   (- (+ (* (log t) (- a 0.5)) (log (* y z))) t)
   (- (* a (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (log(z) <= 106.0) {
		tmp = ((log(t) * (a - 0.5)) + log((y * z))) - t;
	} else {
		tmp = (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 (log(z) <= 106.0d0) then
        tmp = ((log(t) * (a - 0.5d0)) + log((y * z))) - t
    else
        tmp = (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 (Math.log(z) <= 106.0) {
		tmp = ((Math.log(t) * (a - 0.5)) + Math.log((y * z))) - t;
	} else {
		tmp = (a * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if math.log(z) <= 106.0:
		tmp = ((math.log(t) * (a - 0.5)) + math.log((y * z))) - t
	else:
		tmp = (a * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (log(z) <= 106.0)
		tmp = Float64(Float64(Float64(log(t) * Float64(a - 0.5)) + log(Float64(y * z))) - t);
	else
		tmp = Float64(Float64(a * log(t)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (log(z) <= 106.0)
		tmp = ((log(t) * (a - 0.5)) + log((y * z))) - t;
	else
		tmp = (a * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[Log[z], $MachinePrecision], 106.0], N[(N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 z) < 106

   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. fma-def 99.5%

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

      5. remove-double-neg 99.5%

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

      6. remove-double-neg 99.5%

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

      7. sub-neg 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 62.8%

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

   5. Taylor expanded in z around inf 62.8%

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

      1. mul-1-neg 62.8%

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

      2. log-rec 62.8%

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

      3. remove-double-neg 62.8%

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

      4. log-prod 56.2%

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

   7. Simplified 56.2%

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

   if 106 < (log.f64 z)

   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+ 99.8%

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

      3. +-commutative 99.8%

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

      4. fma-def 99.8%

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

      5. remove-double-neg 99.8%

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

      6. remove-double-neg 99.8%

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

      7. sub-neg 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 62.9%

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

   5. Taylor expanded in a around inf 77.9%

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

      1. *-commutative 77.9%

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

   7. Simplified 77.9%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z \leq 106:\\ \;\;\;\;\left(\log t \cdot \left(a - 0.5\right) + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 3: 68.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log z + \log y\\ \mathbf{if}\;t \leq 0.0027:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 - t\right) + \frac{\log t}{\frac{1}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log z) (log y))))
   (if (<= t 0.0027)
     (+ (* (log t) (- a 0.5)) t_1)
     (+ (- t_1 t) (/ (log t) (/ 1.0 a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(z) + log(y);
	double tmp;
	if (t <= 0.0027) {
		tmp = (log(t) * (a - 0.5)) + t_1;
	} else {
		tmp = (t_1 - t) + (log(t) / (1.0 / a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(z) + log(y)
    if (t <= 0.0027d0) then
        tmp = (log(t) * (a - 0.5d0)) + t_1
    else
        tmp = (t_1 - t) + (log(t) / (1.0d0 / a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log(z) + Math.log(y);
	double tmp;
	if (t <= 0.0027) {
		tmp = (Math.log(t) * (a - 0.5)) + t_1;
	} else {
		tmp = (t_1 - t) + (Math.log(t) / (1.0 / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(z) + math.log(y)
	tmp = 0
	if t <= 0.0027:
		tmp = (math.log(t) * (a - 0.5)) + t_1
	else:
		tmp = (t_1 - t) + (math.log(t) / (1.0 / a))
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(z) + log(y))
	tmp = 0.0
	if (t <= 0.0027)
		tmp = Float64(Float64(log(t) * Float64(a - 0.5)) + t_1);
	else
		tmp = Float64(Float64(t_1 - t) + Float64(log(t) / Float64(1.0 / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(z) + log(y);
	tmp = 0.0;
	if (t <= 0.0027)
		tmp = (log(t) * (a - 0.5)) + t_1;
	else
		tmp = (t_1 - t) + (log(t) / (1.0 / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 0.0027], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(t$95$1 - t), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] / N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 0.0027000000000000001

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      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. fma-def 99.3%

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

      5. remove-double-neg 99.3%

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

      6. remove-double-neg 99.3%

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

      7. sub-neg 99.3%

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

      8. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 56.5%

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

   5. Taylor expanded in t around 0 56.3%

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

   if 0.0027000000000000001 < 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. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. flip-+ 76.4%

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

      3. associate-*r/ 75.7%

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

      4. fma-neg 75.7%

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

      5. metadata-eval 75.7%

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

      6. metadata-eval 75.7%

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

      7. sub-neg 75.7%

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

      8. metadata-eval 75.7%

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

   5. Applied egg-rr 75.7%

      \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\frac{\log t \cdot \mathsf{fma}\left(a, a, -0.25\right)}{a + 0.5}} \]
   6. Step-by-step derivation

      1. associate-/l* 76.4%

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

   7. Simplified 76.4%

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

   8. Taylor expanded in a around inf 98.6%

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

   9. Taylor expanded in x around 0 68.6%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.0027:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \left(\log z + \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log z + \log y\right) - t\right) + \frac{\log t}{\frac{1}{a}}\\ \end{array} \]

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

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(log(Float64(x + y)) + 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[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. sub-neg 99.6%

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

   3. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 5: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \left(\log z + \log y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2.1e-5)
   (+ (* (log t) (- a 0.5)) (+ (log z) (log y)))
   (- (* a (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.1e-5) {
		tmp = (log(t) * (a - 0.5)) + (log(z) + log(y));
	} else {
		tmp = (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 <= 2.1d-5) then
        tmp = (log(t) * (a - 0.5d0)) + (log(z) + log(y))
    else
        tmp = (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 <= 2.1e-5) {
		tmp = (Math.log(t) * (a - 0.5)) + (Math.log(z) + Math.log(y));
	} else {
		tmp = (a * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 2.1e-5:
		tmp = (math.log(t) * (a - 0.5)) + (math.log(z) + math.log(y))
	else:
		tmp = (a * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.1e-5)
		tmp = Float64(Float64(log(t) * Float64(a - 0.5)) + Float64(log(z) + log(y)));
	else
		tmp = Float64(Float64(a * log(t)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 2.1e-5)
		tmp = (log(t) * (a - 0.5)) + (log(z) + log(y));
	else
		tmp = (a * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.1e-5], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.09999999999999988e-5

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      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. fma-def 99.3%

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

      5. remove-double-neg 99.3%

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

      6. remove-double-neg 99.3%

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

      7. sub-neg 99.3%

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

      8. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 56.2%

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

   5. Taylor expanded in t around 0 56.0%

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

   if 2.09999999999999988e-5 < 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. fma-def 99.9%

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

      5. remove-double-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 69.4%

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

   5. Taylor expanded in a around inf 98.4%

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

      1. *-commutative 98.4%

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

   7. Simplified 98.4%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \left(\log z + \log y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 6: 69.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. 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. fma-def 99.6%

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

   5. remove-double-neg 99.6%

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

   6. remove-double-neg 99.6%

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

   7. sub-neg 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 62.8%

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

5. Final simplification 62.8%

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

Alternative 7: 73.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\log \left(y \cdot z\right) + -0.5 \cdot \log t\right) - t\\ t_2 := a \cdot \log t - t\\ \mathbf{if}\;a \leq -6.6 \cdot 10^{-26}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -6.4 \cdot 10^{-192}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.6 \cdot 10^{-268}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{elif}\;a \leq 0.00078:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (+ (log (* y z)) (* -0.5 (log t))) t))
        (t_2 (- (* a (log t)) t)))
   (if (<= a -6.6e-26)
     t_2
     (if (<= a -6.4e-192)
       t_1
       (if (<= a -1.6e-268)
         (- (+ (log z) (log y)) t)
         (if (<= a 0.00078) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (log((y * z)) + (-0.5 * log(t))) - t;
	double t_2 = (a * log(t)) - t;
	double tmp;
	if (a <= -6.6e-26) {
		tmp = t_2;
	} else if (a <= -6.4e-192) {
		tmp = t_1;
	} else if (a <= -1.6e-268) {
		tmp = (log(z) + log(y)) - t;
	} else if (a <= 0.00078) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (log((y * z)) + ((-0.5d0) * log(t))) - t
    t_2 = (a * log(t)) - t
    if (a <= (-6.6d-26)) then
        tmp = t_2
    else if (a <= (-6.4d-192)) then
        tmp = t_1
    else if (a <= (-1.6d-268)) then
        tmp = (log(z) + log(y)) - t
    else if (a <= 0.00078d0) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (Math.log((y * z)) + (-0.5 * Math.log(t))) - t;
	double t_2 = (a * Math.log(t)) - t;
	double tmp;
	if (a <= -6.6e-26) {
		tmp = t_2;
	} else if (a <= -6.4e-192) {
		tmp = t_1;
	} else if (a <= -1.6e-268) {
		tmp = (Math.log(z) + Math.log(y)) - t;
	} else if (a <= 0.00078) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (math.log((y * z)) + (-0.5 * math.log(t))) - t
	t_2 = (a * math.log(t)) - t
	tmp = 0
	if a <= -6.6e-26:
		tmp = t_2
	elif a <= -6.4e-192:
		tmp = t_1
	elif a <= -1.6e-268:
		tmp = (math.log(z) + math.log(y)) - t
	elif a <= 0.00078:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(log(Float64(y * z)) + Float64(-0.5 * log(t))) - t)
	t_2 = Float64(Float64(a * log(t)) - t)
	tmp = 0.0
	if (a <= -6.6e-26)
		tmp = t_2;
	elseif (a <= -6.4e-192)
		tmp = t_1;
	elseif (a <= -1.6e-268)
		tmp = Float64(Float64(log(z) + log(y)) - t);
	elseif (a <= 0.00078)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (log((y * z)) + (-0.5 * log(t))) - t;
	t_2 = (a * log(t)) - t;
	tmp = 0.0;
	if (a <= -6.6e-26)
		tmp = t_2;
	elseif (a <= -6.4e-192)
		tmp = t_1;
	elseif (a <= -1.6e-268)
		tmp = (log(z) + log(y)) - t;
	elseif (a <= 0.00078)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[a, -6.6e-26], t$95$2, If[LessEqual[a, -6.4e-192], t$95$1, If[LessEqual[a, -1.6e-268], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 0.00078], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.5999999999999997e-26 or 7.79999999999999986e-4 < 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. fma-def 99.8%

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

      5. remove-double-neg 99.8%

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

      6. remove-double-neg 99.8%

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

      7. sub-neg 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 67.9%

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

   5. Taylor expanded in a around inf 97.5%

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

      1. *-commutative 97.5%

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

   7. Simplified 97.5%

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

   if -6.5999999999999997e-26 < a < -6.4000000000000003e-192 or -1.5999999999999999e-268 < a < 7.79999999999999986e-4

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. associate-+l+ 99.4%

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

      3. +-commutative 99.4%

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

      4. fma-def 99.4%

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

      5. remove-double-neg 99.4%

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

      6. remove-double-neg 99.4%

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

      7. sub-neg 99.4%

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

      8. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 54.8%

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

   5. Taylor expanded in a around 0 54.7%

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

      1. associate-+r+ 54.7%

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

      2. +-commutative 54.7%

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

      3. log-prod 43.6%

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

      4. +-commutative 43.6%

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

      5. *-commutative 43.6%

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

   7. Simplified 43.6%

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

   if -6.4000000000000003e-192 < a < -1.5999999999999999e-268

   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. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. flip-+ 99.9%

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

      3. associate-*r/ 99.9%

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

      4. fma-neg 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

      \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\frac{\log t \cdot \mathsf{fma}\left(a, a, -0.25\right)}{a + 0.5}} \]
   6. Step-by-step derivation

      1. associate-/l* 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in a around inf 89.1%

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

   9. Taylor expanded in x around 0 72.8%

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

   10. Taylor expanded in a around 0 72.8%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.6 \cdot 10^{-26}:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;a \leq -6.4 \cdot 10^{-192}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + -0.5 \cdot \log t\right) - t\\ \mathbf{elif}\;a \leq -1.6 \cdot 10^{-268}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{elif}\;a \leq 0.00078:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + -0.5 \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 8: 71.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(z \cdot \left(y \cdot {t}^{-0.5}\right)\right) - t\\ t_2 := a \cdot \log t - t\\ \mathbf{if}\;a \leq -4 \cdot 10^{-18}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -4.8 \cdot 10^{-186}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -3.5 \cdot 10^{-230}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{elif}\;a \leq 0.00315:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (log (* z (* y (pow t -0.5)))) t)) (t_2 (- (* a (log t)) t)))
   (if (<= a -4e-18)
     t_2
     (if (<= a -4.8e-186)
       t_1
       (if (<= a -3.5e-230)
         (- (+ (log z) (log y)) t)
         (if (<= a 0.00315) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((z * (y * pow(t, -0.5)))) - t;
	double t_2 = (a * log(t)) - t;
	double tmp;
	if (a <= -4e-18) {
		tmp = t_2;
	} else if (a <= -4.8e-186) {
		tmp = t_1;
	} else if (a <= -3.5e-230) {
		tmp = (log(z) + log(y)) - t;
	} else if (a <= 0.00315) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = log((z * (y * (t ** (-0.5d0))))) - t
    t_2 = (a * log(t)) - t
    if (a <= (-4d-18)) then
        tmp = t_2
    else if (a <= (-4.8d-186)) then
        tmp = t_1
    else if (a <= (-3.5d-230)) then
        tmp = (log(z) + log(y)) - t
    else if (a <= 0.00315d0) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log((z * (y * Math.pow(t, -0.5)))) - t;
	double t_2 = (a * Math.log(t)) - t;
	double tmp;
	if (a <= -4e-18) {
		tmp = t_2;
	} else if (a <= -4.8e-186) {
		tmp = t_1;
	} else if (a <= -3.5e-230) {
		tmp = (Math.log(z) + Math.log(y)) - t;
	} else if (a <= 0.00315) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log((z * (y * math.pow(t, -0.5)))) - t
	t_2 = (a * math.log(t)) - t
	tmp = 0
	if a <= -4e-18:
		tmp = t_2
	elif a <= -4.8e-186:
		tmp = t_1
	elif a <= -3.5e-230:
		tmp = (math.log(z) + math.log(y)) - t
	elif a <= 0.00315:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(z * Float64(y * (t ^ -0.5)))) - t)
	t_2 = Float64(Float64(a * log(t)) - t)
	tmp = 0.0
	if (a <= -4e-18)
		tmp = t_2;
	elseif (a <= -4.8e-186)
		tmp = t_1;
	elseif (a <= -3.5e-230)
		tmp = Float64(Float64(log(z) + log(y)) - t);
	elseif (a <= 0.00315)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log((z * (y * (t ^ -0.5)))) - t;
	t_2 = (a * log(t)) - t;
	tmp = 0.0;
	if (a <= -4e-18)
		tmp = t_2;
	elseif (a <= -4.8e-186)
		tmp = t_1;
	elseif (a <= -3.5e-230)
		tmp = (log(z) + log(y)) - t;
	elseif (a <= 0.00315)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(z * N[(y * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[a, -4e-18], t$95$2, If[LessEqual[a, -4.8e-186], t$95$1, If[LessEqual[a, -3.5e-230], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 0.00315], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.0000000000000003e-18 or 0.00315 < 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. fma-def 99.8%

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

      5. remove-double-neg 99.8%

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

      6. remove-double-neg 99.8%

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

      7. sub-neg 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 68.1%

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

   5. Taylor expanded in a around inf 97.5%

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

      1. *-commutative 97.5%

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

   7. Simplified 97.5%

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

   if -4.0000000000000003e-18 < a < -4.80000000000000006e-186 or -3.49999999999999988e-230 < a < 0.00315

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. associate-+l+ 99.4%

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

      3. +-commutative 99.4%

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

      4. fma-def 99.4%

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

      5. remove-double-neg 99.4%

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

      6. remove-double-neg 99.4%

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

      7. sub-neg 99.4%

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

      8. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 54.3%

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

   5. Taylor expanded in a around 0 54.1%

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

   6. Taylor expanded in z around inf 54.1%

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

      1. mul-1-neg 54.1%

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

      2. log-rec 54.1%

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

      3. remove-double-neg 54.1%

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

      4. +-commutative 54.1%

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

      5. *-commutative 54.1%

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

      6. fma-udef 54.1%

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

      7. rem-log-exp 42.8%

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

      8. exp-sum 39.4%

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

      9. rem-exp-log 39.4%

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

      10. fma-udef 39.4%

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

      11. exp-sum 39.4%

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

      12. *-lft-identity 39.4%

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

      13. *-commutative 39.4%

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

      14. exp-to-pow 39.7%

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

      15. unpow1 39.7%

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

      16. *-commutative 39.7%

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

      17. exp-to-pow 39.7%

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

   8. Simplified 39.7%

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

   if -4.80000000000000006e-186 < a < -3.49999999999999988e-230

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. flip-+ 100.0%

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

      3. associate-*r/ 100.0%

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

      4. fma-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\frac{\log t \cdot \mathsf{fma}\left(a, a, -0.25\right)}{a + 0.5}} \]
   6. Step-by-step derivation

      1. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in a around inf 91.1%

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

   9. Taylor expanded in x around 0 80.0%

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

   10. Taylor expanded in a around 0 80.0%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-18}:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;a \leq -4.8 \cdot 10^{-186}:\\ \;\;\;\;\log \left(z \cdot \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{elif}\;a \leq -3.5 \cdot 10^{-230}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{elif}\;a \leq 0.00315:\\ \;\;\;\;\log \left(z \cdot \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 9: 87.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-5}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \log \left(z \cdot \left(x + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.8e-5)
   (+ (* (+ a -0.5) (log t)) (log (* z (+ x y))))
   (- (* a (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.8e-5) {
		tmp = ((a + -0.5) * log(t)) + log((z * (x + y)));
	} else {
		tmp = (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 <= 1.8d-5) then
        tmp = ((a + (-0.5d0)) * log(t)) + log((z * (x + y)))
    else
        tmp = (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 <= 1.8e-5) {
		tmp = ((a + -0.5) * Math.log(t)) + Math.log((z * (x + y)));
	} else {
		tmp = (a * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.8e-5:
		tmp = ((a + -0.5) * math.log(t)) + math.log((z * (x + y)))
	else:
		tmp = (a * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.8e-5)
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) + log(Float64(z * Float64(x + y))));
	else
		tmp = Float64(Float64(a * log(t)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 1.8e-5)
		tmp = ((a + -0.5) * log(t)) + log((z * (x + y)));
	else
		tmp = (a * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.8e-5], 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[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.80000000000000005e-5

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in t around 0 98.5%

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

      1. +-commutative 98.5%

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

      2. log-prod 76.2%

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

      3. +-commutative 76.2%

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

   6. Simplified 76.2%

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

   if 1.80000000000000005e-5 < 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. fma-def 99.9%

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

      5. remove-double-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 69.4%

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

   5. Taylor expanded in a around inf 98.4%

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

      1. *-commutative 98.4%

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

   7. Simplified 98.4%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-5}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \log \left(z \cdot \left(x + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 10: 70.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1250 \lor \neg \left(a \leq 1.9\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1250.0) (not (<= a 1.9)))
   (- (* a (log t)) t)
   (- (+ (log z) (log y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1250.0) || !(a <= 1.9)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = (log(z) + log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-1250.0d0)) .or. (.not. (a <= 1.9d0))) then
        tmp = (a * log(t)) - t
    else
        tmp = (log(z) + log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1250.0) || !(a <= 1.9)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = (Math.log(z) + Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -1250.0) or not (a <= 1.9):
		tmp = (a * math.log(t)) - t
	else:
		tmp = (math.log(z) + math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1250.0) || !(a <= 1.9))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(Float64(log(z) + log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -1250.0) || ~((a <= 1.9)))
		tmp = (a * log(t)) - t;
	else
		tmp = (log(z) + log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1250.0], N[Not[LessEqual[a, 1.9]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1250 or 1.8999999999999999 < 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. fma-def 99.7%

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

      5. remove-double-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 70.9%

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

   5. Taylor expanded in a around inf 98.7%

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

      1. *-commutative 98.7%

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

   7. Simplified 98.7%

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

   if -1250 < a < 1.8999999999999999

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. sub-neg 99.5%

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

      3. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. flip-+ 99.5%

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

      3. associate-*r/ 99.5%

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

      4. fma-neg 99.5%

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

      5. metadata-eval 99.5%

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

      6. metadata-eval 99.5%

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

      7. sub-neg 99.5%

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

      8. metadata-eval 99.5%

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

   5. Applied egg-rr 99.5%

      \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\frac{\log t \cdot \mathsf{fma}\left(a, a, -0.25\right)}{a + 0.5}} \]
   6. Step-by-step derivation

      1. associate-/l* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in a around inf 56.8%

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

   9. Taylor expanded in x around 0 37.7%

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

   10. Taylor expanded in a around 0 37.7%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1250 \lor \neg \left(a \leq 1.9\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \]

Alternative 11: 77.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;t \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\log \left(x + y\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= t 2.1e-5) (+ (log (+ x y)) t_1) (- t_1 t))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if (t <= 2.1e-5) {
		tmp = log((x + y)) + t_1;
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if (t <= 2.1d-5) then
        tmp = log((x + y)) + t_1
    else
        tmp = t_1 - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if (t <= 2.1e-5) {
		tmp = Math.log((x + y)) + t_1;
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if t <= 2.1e-5:
		tmp = math.log((x + y)) + t_1
	else:
		tmp = t_1 - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (t <= 2.1e-5)
		tmp = Float64(log(Float64(x + y)) + t_1);
	else
		tmp = Float64(t_1 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if (t <= 2.1e-5)
		tmp = log((x + y)) + t_1;
	else
		tmp = t_1 - t;
	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[t, 2.1e-5], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision], N[(t$95$1 - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.09999999999999988e-5

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      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. fma-def 99.3%

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

      5. remove-double-neg 99.3%

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

      6. remove-double-neg 99.3%

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

      7. sub-neg 99.3%

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

      8. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in a around inf 55.7%

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

      1. *-commutative 55.7%

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

   6. Simplified 55.7%

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

   if 2.09999999999999988e-5 < 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. fma-def 99.9%

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

      5. remove-double-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 69.4%

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

   5. Taylor expanded in a around inf 98.4%

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

      1. *-commutative 98.4%

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

   7. Simplified 98.4%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\log \left(x + y\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 12: 69.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1250 \lor \neg \left(a \leq 6.5 \cdot 10^{-13}\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1250.0) (not (<= a 6.5e-13)))
   (- (* a (log t)) t)
   (- (log y) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1250.0) || !(a <= 6.5e-13)) {
		tmp = (a * log(t)) - 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 <= (-1250.0d0)) .or. (.not. (a <= 6.5d-13))) then
        tmp = (a * log(t)) - 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 <= -1250.0) || !(a <= 6.5e-13)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = Math.log(y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -1250.0) or not (a <= 6.5e-13):
		tmp = (a * math.log(t)) - t
	else:
		tmp = math.log(y) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1250.0) || !(a <= 6.5e-13))
		tmp = Float64(Float64(a * log(t)) - 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 <= -1250.0) || ~((a <= 6.5e-13)))
		tmp = (a * log(t)) - t;
	else
		tmp = log(y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1250.0], N[Not[LessEqual[a, 6.5e-13]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1250 or 6.49999999999999957e-13 < 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. fma-def 99.7%

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

      5. remove-double-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 71.1%

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

   5. Taylor expanded in a around inf 98.0%

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

      1. *-commutative 98.0%

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

   7. Simplified 98.0%

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

   if -1250 < a < 6.49999999999999957e-13

   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. fma-def 99.5%

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

      5. remove-double-neg 99.5%

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

      6. remove-double-neg 99.5%

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

      7. sub-neg 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in t around inf 56.1%

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

      1. neg-mul-1 56.1%

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

   6. Simplified 56.1%

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

   7. Taylor expanded in x around 0 37.4%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1250 \lor \neg \left(a \leq 6.5 \cdot 10^{-13}\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log y - t\\ \end{array} \]

Alternative 13: 57.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -29000000000 \lor \neg \left(a \leq 6.5 \cdot 10^{-13}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -29000000000.0) (not (<= a 6.5e-13)))
   (* a (log t))
   (- (log y) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -29000000000.0) || !(a <= 6.5e-13)) {
		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 <= (-29000000000.0d0)) .or. (.not. (a <= 6.5d-13))) 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 <= -29000000000.0) || !(a <= 6.5e-13)) {
		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 <= -29000000000.0) or not (a <= 6.5e-13):
		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 <= -29000000000.0) || !(a <= 6.5e-13))
		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 <= -29000000000.0) || ~((a <= 6.5e-13)))
		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, -29000000000.0], N[Not[LessEqual[a, 6.5e-13]], $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 < -2.9e10 or 6.49999999999999957e-13 < 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. sub-neg 99.7%

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

      3. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. flip-+ 58.4%

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

      3. associate-*r/ 56.8%

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

      4. fma-neg 56.8%

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

      5. metadata-eval 56.8%

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

      6. metadata-eval 56.8%

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

      7. sub-neg 56.8%

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

      8. metadata-eval 56.8%

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

   5. Applied egg-rr 56.8%

      \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\frac{\log t \cdot \mathsf{fma}\left(a, a, -0.25\right)}{a + 0.5}} \]
   6. Step-by-step derivation

      1. associate-/l* 58.3%

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

   7. Simplified 58.3%

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

   8. Taylor expanded in a around inf 98.4%

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

   9. Taylor expanded in x around 0 70.3%

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

   10. Taylor expanded in a around inf 76.0%

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

   if -2.9e10 < a < 6.49999999999999957e-13

   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. fma-def 99.5%

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

      5. remove-double-neg 99.5%

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

      6. remove-double-neg 99.5%

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

      7. sub-neg 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in t around inf 56.1%

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

      1. neg-mul-1 56.1%

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

   6. Simplified 56.1%

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

   7. Taylor expanded in x around 0 37.6%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -29000000000 \lor \neg \left(a \leq 6.5 \cdot 10^{-13}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log y - t\\ \end{array} \]

Alternative 14: 41.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.0025:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (if (<= t 0.0025) (log (+ x y)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.0025) {
		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 <= 0.0025d0) 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 <= 0.0025) {
		tmp = Math.log((x + y));
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 0.0025:
		tmp = math.log((x + y))
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 0.0025)
		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 <= 0.0025)
		tmp = log((x + y));
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 0.0025], N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 0.00250000000000000005

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      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. fma-def 99.3%

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

      5. remove-double-neg 99.3%

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

      6. remove-double-neg 99.3%

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

      7. sub-neg 99.3%

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

      8. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in t around inf 9.8%

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

      1. neg-mul-1 9.8%

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

   6. Simplified 9.8%

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

   7. Taylor expanded in t around 0 9.8%

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

   if 0.00250000000000000005 < 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. fma-def 99.9%

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

      5. remove-double-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 69.2%

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

   5. Taylor expanded in t around inf 70.8%

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

      1. neg-mul-1 70.8%

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

   7. Simplified 70.8%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.0025:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 15: 62.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{+35}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (if (<= t 8e+35) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 8e+35) {
		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 (t <= 8d+35) 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 (t <= 8e+35) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 8e+35:
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 8e+35)
		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 (t <= 8e+35)
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 8e+35], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 7.9999999999999997e35

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. sub-neg 99.4%

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

      3. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. flip-+ 80.2%

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

      3. associate-*r/ 79.5%

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

      4. fma-neg 79.5%

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

      5. metadata-eval 79.5%

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

      6. metadata-eval 79.5%

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

      7. sub-neg 79.5%

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

      8. metadata-eval 79.5%

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

   5. Applied egg-rr 79.5%

      \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\frac{\log t \cdot \mathsf{fma}\left(a, a, -0.25\right)}{a + 0.5}} \]
   6. Step-by-step derivation

      1. associate-/l* 80.2%

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

   7. Simplified 80.2%

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

   8. Taylor expanded in a around inf 60.5%

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

   9. Taylor expanded in x around 0 43.1%

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

   10. Taylor expanded in a around inf 51.2%

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

   if 7.9999999999999997e35 < 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. fma-def 99.9%

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

      5. remove-double-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 68.4%

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

   5. Taylor expanded in t around inf 77.6%

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

      1. neg-mul-1 77.6%

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

   7. Simplified 77.6%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{+35}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 16: 37.9% 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. fma-def 99.6%

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

   5. remove-double-neg 99.6%

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

   6. remove-double-neg 99.6%

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

   7. sub-neg 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 62.8%

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

5. Taylor expanded in t around inf 36.5%

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

   1. neg-mul-1 36.5%

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

7. Simplified 36.5%

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

8. Final simplification 36.5%

   \[\leadsto -t \]

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 2023185 
(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))))