Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 23.9s

Alternatives: 17

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

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

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 99.7%

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

   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. Final simplification 99.7%

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

Alternative 2: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a - 0.5 \leq -400000000:\\ \;\;\;\;t_1 - t\\ \mathbf{elif}\;a - 0.5 \leq -0.49999999998:\\ \;\;\;\;\log y + \left(\log z - \left(t - -0.5 \cdot \log t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= (- a 0.5) -400000000.0)
     (- t_1 t)
     (if (<= (- a 0.5) -0.49999999998)
       (+ (log y) (- (log z) (- t (* -0.5 (log t)))))
       (+ (- (log z) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if ((a - 0.5) <= -400000000.0) {
		tmp = t_1 - t;
	} else if ((a - 0.5) <= -0.49999999998) {
		tmp = log(y) + (log(z) - (t - (-0.5 * log(t))));
	} else {
		tmp = (log(z) - t) + t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if ((a - 0.5d0) <= (-400000000.0d0)) then
        tmp = t_1 - t
    else if ((a - 0.5d0) <= (-0.49999999998d0)) then
        tmp = log(y) + (log(z) - (t - ((-0.5d0) * log(t))))
    else
        tmp = (log(z) - t) + t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if ((a - 0.5) <= -400000000.0) {
		tmp = t_1 - t;
	} else if ((a - 0.5) <= -0.49999999998) {
		tmp = Math.log(y) + (Math.log(z) - (t - (-0.5 * Math.log(t))));
	} else {
		tmp = (Math.log(z) - t) + t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if (a - 0.5) <= -400000000.0:
		tmp = t_1 - t
	elif (a - 0.5) <= -0.49999999998:
		tmp = math.log(y) + (math.log(z) - (t - (-0.5 * math.log(t))))
	else:
		tmp = (math.log(z) - t) + t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (Float64(a - 0.5) <= -400000000.0)
		tmp = Float64(t_1 - t);
	elseif (Float64(a - 0.5) <= -0.49999999998)
		tmp = Float64(log(y) + Float64(log(z) - Float64(t - Float64(-0.5 * log(t)))));
	else
		tmp = Float64(Float64(log(z) - t) + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if ((a - 0.5) <= -400000000.0)
		tmp = t_1 - t;
	elseif ((a - 0.5) <= -0.49999999998)
		tmp = log(y) + (log(z) - (t - (-0.5 * log(t))));
	else
		tmp = (log(z) - t) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -400000000.0], N[(t$95$1 - t), $MachinePrecision], If[LessEqual[N[(a - 0.5), $MachinePrecision], -0.49999999998], N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - N[(t - N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 a 1/2) < -4e8

   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 73.3%

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

      1. associate--l+ 73.3%

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

      2. remove-double-neg 73.3%

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

      3. log-rec 73.3%

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

      4. mul-1-neg 73.3%

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

      5. associate--l+ 73.3%

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

      6. fma-def 73.3%

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

      7. sub-neg 73.3%

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

      8. metadata-eval 73.3%

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

      9. +-commutative 73.3%

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

      10. mul-1-neg 73.3%

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

      11. log-rec 73.3%

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

      12. remove-double-neg 73.3%

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

   6. Simplified 73.3%

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

   7. Taylor expanded in a around inf 99.0%

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

      1. *-commutative 99.0%

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

   9. Simplified 99.0%

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

   if -4e8 < (-.f64 a 1/2) < -0.49999999998

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. fma-def 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 70.0%

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

   5. Taylor expanded in a around 0 69.2%

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

      1. associate-+r+ 69.2%

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

      2. +-commutative 69.2%

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

      3. log-prod 48.4%

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

   7. Simplified 48.4%

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

      1. associate--l+ 48.4%

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

      2. log-prod 69.2%

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

      3. associate-+l+ 69.3%

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

   9. Applied egg-rr 69.3%

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

   if -0.49999999998 < (-.f64 a 1/2)

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 74.0%

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

   5. Taylor expanded in a around inf 98.7%

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

      1. *-commutative 68.3%

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

   7. Simplified 98.7%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -400000000:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;a - 0.5 \leq -0.49999999998:\\ \;\;\;\;\log y + \left(\log z - \left(t - -0.5 \cdot \log t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 3: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a - 0.5 \leq -400000000:\\ \;\;\;\;t_1 - t\\ \mathbf{elif}\;a - 0.5 \leq -0.49999999998:\\ \;\;\;\;\left(\log z + \log \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= (- a 0.5) -400000000.0)
     (- t_1 t)
     (if (<= (- a 0.5) -0.49999999998)
       (- (+ (log z) (log (* y (pow t -0.5)))) t)
       (+ (- (log z) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if ((a - 0.5) <= -400000000.0) {
		tmp = t_1 - t;
	} else if ((a - 0.5) <= -0.49999999998) {
		tmp = (log(z) + log((y * pow(t, -0.5)))) - t;
	} else {
		tmp = (log(z) - t) + t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if ((a - 0.5d0) <= (-400000000.0d0)) then
        tmp = t_1 - t
    else if ((a - 0.5d0) <= (-0.49999999998d0)) then
        tmp = (log(z) + log((y * (t ** (-0.5d0))))) - t
    else
        tmp = (log(z) - t) + t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if ((a - 0.5) <= -400000000.0) {
		tmp = t_1 - t;
	} else if ((a - 0.5) <= -0.49999999998) {
		tmp = (Math.log(z) + Math.log((y * Math.pow(t, -0.5)))) - t;
	} else {
		tmp = (Math.log(z) - t) + t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if (a - 0.5) <= -400000000.0:
		tmp = t_1 - t
	elif (a - 0.5) <= -0.49999999998:
		tmp = (math.log(z) + math.log((y * math.pow(t, -0.5)))) - t
	else:
		tmp = (math.log(z) - t) + t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (Float64(a - 0.5) <= -400000000.0)
		tmp = Float64(t_1 - t);
	elseif (Float64(a - 0.5) <= -0.49999999998)
		tmp = Float64(Float64(log(z) + log(Float64(y * (t ^ -0.5)))) - t);
	else
		tmp = Float64(Float64(log(z) - t) + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if ((a - 0.5) <= -400000000.0)
		tmp = t_1 - t;
	elseif ((a - 0.5) <= -0.49999999998)
		tmp = (log(z) + log((y * (t ^ -0.5)))) - t;
	else
		tmp = (log(z) - t) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -400000000.0], N[(t$95$1 - t), $MachinePrecision], If[LessEqual[N[(a - 0.5), $MachinePrecision], -0.49999999998], N[(N[(N[Log[z], $MachinePrecision] + N[Log[N[(y * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 a 1/2) < -4e8

   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 73.3%

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

      1. associate--l+ 73.3%

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

      2. remove-double-neg 73.3%

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

      3. log-rec 73.3%

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

      4. mul-1-neg 73.3%

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

      5. associate--l+ 73.3%

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

      6. fma-def 73.3%

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

      7. sub-neg 73.3%

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

      8. metadata-eval 73.3%

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

      9. +-commutative 73.3%

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

      10. mul-1-neg 73.3%

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

      11. log-rec 73.3%

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

      12. remove-double-neg 73.3%

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

   6. Simplified 73.3%

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

   7. Taylor expanded in a around inf 99.0%

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

      1. *-commutative 99.0%

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

   9. Simplified 99.0%

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

   if -4e8 < (-.f64 a 1/2) < -0.49999999998

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. fma-def 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 70.0%

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

   5. Taylor expanded in a around 0 69.2%

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

   6. Taylor expanded in y around inf 69.2%

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

      1. mul-1-neg 69.2%

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

      2. log-rec 69.2%

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

      3. remove-double-neg 69.2%

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

      4. log-pow 69.2%

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

      5. log-prod 63.5%

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

   8. Simplified 63.5%

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

   if -0.49999999998 < (-.f64 a 1/2)

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 74.0%

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

   5. Taylor expanded in a around inf 98.7%

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

      1. *-commutative 68.3%

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

   7. Simplified 98.7%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -400000000:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;a - 0.5 \leq -0.49999999998:\\ \;\;\;\;\left(\log z + \log \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 4: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log z \leq 50:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(\left(a + -0.5\right) \cdot \log t - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (log z) 50.0)
   (+ (log (* (+ x y) z)) (- (* (+ a -0.5) (log t)) t))
   (+ (- (log z) t) (* a (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (log(z) <= 50.0) {
		tmp = log(((x + y) * z)) + (((a + -0.5) * log(t)) - t);
	} else {
		tmp = (log(z) - t) + (a * log(t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (log(z) <= 50.0d0) then
        tmp = log(((x + y) * z)) + (((a + (-0.5d0)) * log(t)) - t)
    else
        tmp = (log(z) - t) + (a * log(t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (Math.log(z) <= 50.0) {
		tmp = Math.log(((x + y) * z)) + (((a + -0.5) * Math.log(t)) - t);
	} else {
		tmp = (Math.log(z) - t) + (a * Math.log(t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if math.log(z) <= 50.0:
		tmp = math.log(((x + y) * z)) + (((a + -0.5) * math.log(t)) - t)
	else:
		tmp = (math.log(z) - t) + (a * math.log(t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (log(z) <= 50.0)
		tmp = Float64(log(Float64(Float64(x + y) * z)) + Float64(Float64(Float64(a + -0.5) * log(t)) - t));
	else
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (log(z) <= 50.0)
		tmp = log(((x + y) * z)) + (((a + -0.5) * log(t)) - t);
	else
		tmp = (log(z) - t) + (a * log(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[Log[z], $MachinePrecision], 50.0], N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 z) < 50

   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. metadata-eval 99.4%

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

      2. sub-neg 99.4%

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

      3. associate-+r- 99.4%

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

      4. associate-+l- 99.4%

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

      5. +-commutative 99.4%

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

      6. sum-log 91.3%

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

      7. sub-neg 91.3%

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

      8. metadata-eval 91.3%

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

   5. Applied egg-rr 91.3%

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

   if 50 < (log.f64 z)

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-def 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 74.9%

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

   5. Taylor expanded in a around inf 83.7%

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

      1. *-commutative 37.1%

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

   7. Simplified 83.7%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z \leq 50:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(\left(a + -0.5\right) \cdot \log t - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 5: 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 6: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 230:\\ \;\;\;\;\log z + \left(\log t \cdot \left(a - 0.5\right) + \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 230.0)
   (+ (log z) (+ (* (log t) (- a 0.5)) (log y)))
   (+ (- (log z) t) (* a (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 230.0) {
		tmp = log(z) + ((log(t) * (a - 0.5)) + log(y));
	} else {
		tmp = (log(z) - t) + (a * log(t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 230.0d0) then
        tmp = log(z) + ((log(t) * (a - 0.5d0)) + log(y))
    else
        tmp = (log(z) - t) + (a * log(t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 230.0) {
		tmp = Math.log(z) + ((Math.log(t) * (a - 0.5)) + Math.log(y));
	} else {
		tmp = (Math.log(z) - t) + (a * Math.log(t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 230.0:
		tmp = math.log(z) + ((math.log(t) * (a - 0.5)) + math.log(y))
	else:
		tmp = (math.log(z) - t) + (a * math.log(t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 230.0)
		tmp = Float64(log(z) + Float64(Float64(log(t) * Float64(a - 0.5)) + log(y)));
	else
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 230.0)
		tmp = log(z) + ((log(t) * (a - 0.5)) + log(y));
	else
		tmp = (log(z) - t) + (a * log(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 230.0], N[(N[Log[z], $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 230

   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. +-commutative 99.3%

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

      3. associate-+l+ 99.3%

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

      4. +-commutative 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 69.7%

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

   5. Taylor expanded in t around 0 69.3%

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

   if 230 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 73.7%

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

   5. Taylor expanded in a around inf 99.6%

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

      1. *-commutative 27.9%

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

   7. Simplified 99.6%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 230:\\ \;\;\;\;\log z + \left(\log t \cdot \left(a - 0.5\right) + \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 7: 70.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-+l+ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-def 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 71.8%

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

5. Final simplification 71.8%

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

Alternative 8: 75.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ t_2 := \log \left(x + y\right) + t_1\\ \mathbf{if}\;t \leq 4.6 \cdot 10^{-220}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-194}:\\ \;\;\;\;-0.5 \cdot \log t + \log \left(y \cdot z\right)\\ \mathbf{elif}\;t \leq 210:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))) (t_2 (+ (log (+ x y)) t_1)))
   (if (<= t 4.6e-220)
     t_2
     (if (<= t 3.5e-194)
       (+ (* -0.5 (log t)) (log (* y z)))
       (if (<= t 210.0) t_2 (- t_1 t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double t_2 = log((x + y)) + t_1;
	double tmp;
	if (t <= 4.6e-220) {
		tmp = t_2;
	} else if (t <= 3.5e-194) {
		tmp = (-0.5 * log(t)) + log((y * z));
	} else if (t <= 210.0) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = a * log(t)
    t_2 = log((x + y)) + t_1
    if (t <= 4.6d-220) then
        tmp = t_2
    else if (t <= 3.5d-194) then
        tmp = ((-0.5d0) * log(t)) + log((y * z))
    else if (t <= 210.0d0) then
        tmp = t_2
    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 t_2 = Math.log((x + y)) + t_1;
	double tmp;
	if (t <= 4.6e-220) {
		tmp = t_2;
	} else if (t <= 3.5e-194) {
		tmp = (-0.5 * Math.log(t)) + Math.log((y * z));
	} else if (t <= 210.0) {
		tmp = t_2;
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	t_2 = math.log((x + y)) + t_1
	tmp = 0
	if t <= 4.6e-220:
		tmp = t_2
	elif t <= 3.5e-194:
		tmp = (-0.5 * math.log(t)) + math.log((y * z))
	elif t <= 210.0:
		tmp = t_2
	else:
		tmp = t_1 - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	t_2 = Float64(log(Float64(x + y)) + t_1)
	tmp = 0.0
	if (t <= 4.6e-220)
		tmp = t_2;
	elseif (t <= 3.5e-194)
		tmp = Float64(Float64(-0.5 * log(t)) + log(Float64(y * z)));
	elseif (t <= 210.0)
		tmp = t_2;
	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);
	t_2 = log((x + y)) + t_1;
	tmp = 0.0;
	if (t <= 4.6e-220)
		tmp = t_2;
	elseif (t <= 3.5e-194)
		tmp = (-0.5 * log(t)) + log((y * z));
	elseif (t <= 210.0)
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t, 4.6e-220], t$95$2, If[LessEqual[t, 3.5e-194], N[(N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 210.0], t$95$2, N[(t$95$1 - t), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 4.59999999999999961e-220 or 3.5000000000000003e-194 < t < 210

   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+ 99.4%

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

      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 a around inf 58.3%

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

      1. *-commutative 58.3%

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

   6. Simplified 58.3%

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

   if 4.59999999999999961e-220 < t < 3.5000000000000003e-194

   1. Initial program 99.1%

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

      1. associate--l+ 99.1%

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

      2. +-commutative 99.1%

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

      3. associate-+l+ 99.1%

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

      4. +-commutative 99.1%

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

      5. fma-def 99.1%

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

      6. sub-neg 99.1%

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

      7. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around 0 79.1%

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

   5. Taylor expanded in a around 0 56.8%

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

      1. associate-+r+ 56.7%

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

      2. +-commutative 56.7%

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

      3. log-prod 52.1%

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

   7. Simplified 52.1%

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

   8. Taylor expanded in t around 0 52.1%

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

   if 210 < 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 73.7%

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

      1. associate--l+ 73.7%

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

      2. remove-double-neg 73.7%

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

      3. log-rec 73.7%

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

      4. mul-1-neg 73.7%

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

      5. associate--l+ 73.7%

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

      6. fma-def 73.7%

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

      7. sub-neg 73.7%

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

      8. metadata-eval 73.7%

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

      9. +-commutative 73.7%

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

      10. mul-1-neg 73.7%

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

      11. log-rec 73.7%

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

      12. remove-double-neg 73.7%

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

   6. Simplified 73.7%

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

   7. Taylor expanded in a around inf 99.5%

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

      1. *-commutative 99.5%

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

   9. Simplified 99.5%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{-220}:\\ \;\;\;\;\log \left(x + y\right) + a \cdot \log t\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-194}:\\ \;\;\;\;-0.5 \cdot \log t + \log \left(y \cdot z\right)\\ \mathbf{elif}\;t \leq 210:\\ \;\;\;\;\log \left(x + y\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 9: 86.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.0065:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \log \left(\left(x + y\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 0.0065)
   (+ (* (+ a -0.5) (log t)) (log (* (+ x y) z)))
   (+ (- (log z) t) (* a (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.0065) {
		tmp = ((a + -0.5) * log(t)) + log(((x + y) * z));
	} else {
		tmp = (log(z) - t) + (a * log(t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 0.0065d0) then
        tmp = ((a + (-0.5d0)) * log(t)) + log(((x + y) * z))
    else
        tmp = (log(z) - t) + (a * log(t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.0065) {
		tmp = ((a + -0.5) * Math.log(t)) + Math.log(((x + y) * z));
	} else {
		tmp = (Math.log(z) - t) + (a * Math.log(t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 0.0065:
		tmp = ((a + -0.5) * math.log(t)) + math.log(((x + y) * z))
	else:
		tmp = (math.log(z) - t) + (a * math.log(t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 0.0065)
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) + log(Float64(Float64(x + y) * z)));
	else
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 0.0065)
		tmp = ((a + -0.5) * log(t)) + log(((x + y) * z));
	else
		tmp = (log(z) - t) + (a * log(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 0.0065], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 0.0064999999999999997

   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 97.9%

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

      1. +-commutative 97.9%

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

      2. log-prod 68.5%

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

      3. +-commutative 68.5%

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

   6. Simplified 68.5%

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

   if 0.0064999999999999997 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 73.7%

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

   5. Taylor expanded in a around inf 99.6%

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

      1. *-commutative 27.9%

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

   7. Simplified 99.6%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.0065:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \log \left(\left(x + y\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 10: 71.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -105000:\\ \;\;\;\;t_1 - t\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-68}:\\ \;\;\;\;\log \left(z \cdot \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= a -105000.0)
     (- t_1 t)
     (if (<= a 2e-68)
       (- (log (* z (* y (pow t -0.5)))) t)
       (+ (- (log z) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if (a <= -105000.0) {
		tmp = t_1 - t;
	} else if (a <= 2e-68) {
		tmp = log((z * (y * pow(t, -0.5)))) - t;
	} else {
		tmp = (log(z) - t) + t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if (a <= (-105000.0d0)) then
        tmp = t_1 - t
    else if (a <= 2d-68) then
        tmp = log((z * (y * (t ** (-0.5d0))))) - t
    else
        tmp = (log(z) - t) + t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if (a <= -105000.0) {
		tmp = t_1 - t;
	} else if (a <= 2e-68) {
		tmp = Math.log((z * (y * Math.pow(t, -0.5)))) - t;
	} else {
		tmp = (Math.log(z) - t) + t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if a <= -105000.0:
		tmp = t_1 - t
	elif a <= 2e-68:
		tmp = math.log((z * (y * math.pow(t, -0.5)))) - t
	else:
		tmp = (math.log(z) - t) + t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (a <= -105000.0)
		tmp = Float64(t_1 - t);
	elseif (a <= 2e-68)
		tmp = Float64(log(Float64(z * Float64(y * (t ^ -0.5)))) - t);
	else
		tmp = Float64(Float64(log(z) - t) + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if (a <= -105000.0)
		tmp = t_1 - t;
	elseif (a <= 2e-68)
		tmp = log((z * (y * (t ^ -0.5)))) - t;
	else
		tmp = (log(z) - t) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -105000.0], N[(t$95$1 - t), $MachinePrecision], If[LessEqual[a, 2e-68], N[(N[Log[N[(z * N[(y * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -105000

   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 73.3%

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

      1. associate--l+ 73.3%

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

      2. remove-double-neg 73.3%

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

      3. log-rec 73.3%

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

      4. mul-1-neg 73.3%

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

      5. associate--l+ 73.3%

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

      6. fma-def 73.3%

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

      7. sub-neg 73.3%

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

      8. metadata-eval 73.3%

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

      9. +-commutative 73.3%

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

      10. mul-1-neg 73.3%

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

      11. log-rec 73.3%

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

      12. remove-double-neg 73.3%

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

   6. Simplified 73.3%

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

   7. Taylor expanded in a around inf 99.0%

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

      1. *-commutative 99.0%

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

   9. Simplified 99.0%

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

   if -105000 < a < 2.00000000000000013e-68

   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+ 99.6%

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

      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 70.6%

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

      1. associate--l+ 70.6%

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

      2. remove-double-neg 70.6%

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

      3. log-rec 70.6%

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

      4. mul-1-neg 70.6%

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

      5. associate--l+ 70.6%

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

      6. fma-def 70.6%

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

      7. sub-neg 70.6%

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

      8. metadata-eval 70.6%

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

      9. +-commutative 70.6%

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

      10. mul-1-neg 70.6%

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

      11. log-rec 70.6%

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

      12. remove-double-neg 70.6%

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

   6. Simplified 70.6%

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

   7. Taylor expanded in a around 0 69.8%

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

      1. associate-+r+ 69.8%

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

      2. log-prod 48.6%

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

      3. log-pow 48.6%

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

      4. log-prod 42.2%

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

      5. associate-*r* 50.7%

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

   9. Simplified 50.7%

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

   if 2.00000000000000013e-68 < a

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 72.7%

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

   5. Taylor expanded in a around inf 96.7%

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

      1. *-commutative 62.8%

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

   7. Simplified 96.7%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -105000:\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-68}:\\ \;\;\;\;\log \left(z \cdot \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 11: 73.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ t_2 := \log z + t_1\\ \mathbf{if}\;t \leq 3.15 \cdot 10^{-264}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-193}:\\ \;\;\;\;-0.5 \cdot \log t + \log \left(y \cdot z\right)\\ \mathbf{elif}\;t \leq 0.0052:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))) (t_2 (+ (log z) t_1)))
   (if (<= t 3.15e-264)
     t_2
     (if (<= t 1.3e-193)
       (+ (* -0.5 (log t)) (log (* y z)))
       (if (<= t 0.0052) t_2 (- t_1 t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double t_2 = log(z) + t_1;
	double tmp;
	if (t <= 3.15e-264) {
		tmp = t_2;
	} else if (t <= 1.3e-193) {
		tmp = (-0.5 * log(t)) + log((y * z));
	} else if (t <= 0.0052) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = a * log(t)
    t_2 = log(z) + t_1
    if (t <= 3.15d-264) then
        tmp = t_2
    else if (t <= 1.3d-193) then
        tmp = ((-0.5d0) * log(t)) + log((y * z))
    else if (t <= 0.0052d0) then
        tmp = t_2
    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 t_2 = Math.log(z) + t_1;
	double tmp;
	if (t <= 3.15e-264) {
		tmp = t_2;
	} else if (t <= 1.3e-193) {
		tmp = (-0.5 * Math.log(t)) + Math.log((y * z));
	} else if (t <= 0.0052) {
		tmp = t_2;
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	t_2 = math.log(z) + t_1
	tmp = 0
	if t <= 3.15e-264:
		tmp = t_2
	elif t <= 1.3e-193:
		tmp = (-0.5 * math.log(t)) + math.log((y * z))
	elif t <= 0.0052:
		tmp = t_2
	else:
		tmp = t_1 - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	t_2 = Float64(log(z) + t_1)
	tmp = 0.0
	if (t <= 3.15e-264)
		tmp = t_2;
	elseif (t <= 1.3e-193)
		tmp = Float64(Float64(-0.5 * log(t)) + log(Float64(y * z)));
	elseif (t <= 0.0052)
		tmp = t_2;
	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);
	t_2 = log(z) + t_1;
	tmp = 0.0;
	if (t <= 3.15e-264)
		tmp = t_2;
	elseif (t <= 1.3e-193)
		tmp = (-0.5 * log(t)) + log((y * z));
	elseif (t <= 0.0052)
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[Log[z], $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t, 3.15e-264], t$95$2, If[LessEqual[t, 1.3e-193], N[(N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.0052], t$95$2, N[(t$95$1 - t), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.1499999999999999e-264 or 1.30000000000000004e-193 < t < 0.0051999999999999998

   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. +-commutative 99.3%

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

      3. associate-+l+ 99.4%

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

      4. +-commutative 99.4%

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

      5. fma-def 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 69.1%

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

   5. Taylor expanded in t around 0 68.5%

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

   6. Taylor expanded in a around inf 60.1%

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

      1. *-commutative 60.1%

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

   8. Simplified 60.1%

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

   if 3.1499999999999999e-264 < t < 1.30000000000000004e-193

   1. Initial program 99.2%

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

      1. associate--l+ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 99.2%

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

      4. +-commutative 99.2%

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

      5. fma-def 99.2%

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

      6. sub-neg 99.2%

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

      7. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 73.9%

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

   5. Taylor expanded in a around 0 50.3%

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

      1. associate-+r+ 50.2%

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

      2. +-commutative 50.2%

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

      3. log-prod 44.2%

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

   7. Simplified 44.2%

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

   8. Taylor expanded in t around 0 44.2%

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

   if 0.0051999999999999998 < 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 73.1%

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

      1. associate--l+ 73.1%

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

      2. remove-double-neg 73.1%

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

      3. log-rec 73.1%

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

      4. mul-1-neg 73.1%

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

      5. associate--l+ 73.1%

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

      6. fma-def 73.1%

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

      7. sub-neg 73.1%

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

      8. metadata-eval 73.1%

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

      9. +-commutative 73.1%

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

      10. mul-1-neg 73.1%

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

      11. log-rec 73.1%

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

      12. remove-double-neg 73.1%

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

   6. Simplified 73.1%

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

   7. Taylor expanded in a around inf 98.8%

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

      1. *-commutative 98.8%

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

   9. Simplified 98.8%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.15 \cdot 10^{-264}:\\ \;\;\;\;\log z + a \cdot \log t\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-193}:\\ \;\;\;\;-0.5 \cdot \log t + \log \left(y \cdot z\right)\\ \mathbf{elif}\;t \leq 0.0052:\\ \;\;\;\;\log z + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 12: 75.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{-114} \lor \neg \left(a \leq -1.05 \cdot 10^{-175}\right):\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \log t + \log \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -3.2e-114) (not (<= a -1.05e-175)))
   (+ (- (log z) t) (* a (log t)))
   (+ (* -0.5 (log t)) (log (* y z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3.2e-114) || !(a <= -1.05e-175)) {
		tmp = (log(z) - t) + (a * log(t));
	} else {
		tmp = (-0.5 * log(t)) + log((y * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-3.2d-114)) .or. (.not. (a <= (-1.05d-175)))) then
        tmp = (log(z) - t) + (a * log(t))
    else
        tmp = ((-0.5d0) * log(t)) + log((y * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3.2e-114) || !(a <= -1.05e-175)) {
		tmp = (Math.log(z) - t) + (a * Math.log(t));
	} else {
		tmp = (-0.5 * Math.log(t)) + Math.log((y * z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -3.2e-114) or not (a <= -1.05e-175):
		tmp = (math.log(z) - t) + (a * math.log(t))
	else:
		tmp = (-0.5 * math.log(t)) + math.log((y * z))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -3.2e-114) || !(a <= -1.05e-175))
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	else
		tmp = Float64(Float64(-0.5 * log(t)) + log(Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -3.2e-114) || ~((a <= -1.05e-175)))
		tmp = (log(z) - t) + (a * log(t));
	else
		tmp = (-0.5 * log(t)) + log((y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -3.2e-114], N[Not[LessEqual[a, -1.05e-175]], $MachinePrecision]], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.2000000000000002e-114 or -1.05e-175 < a

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 71.1%

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

   5. Taylor expanded in a around inf 81.4%

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

      1. *-commutative 41.8%

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

   7. Simplified 81.4%

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

   if -3.2000000000000002e-114 < a < -1.05e-175

   1. Initial program 99.0%

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

      1. associate--l+ 99.0%

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

      2. +-commutative 99.0%

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

      3. associate-+l+ 99.1%

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

      4. +-commutative 99.1%

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

      5. fma-def 99.1%

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

      6. sub-neg 99.1%

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

      7. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around 0 82.5%

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

   5. Taylor expanded in a around 0 82.6%

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

      1. associate-+r+ 82.5%

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

      2. +-commutative 82.5%

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

      3. log-prod 65.6%

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

   7. Simplified 65.6%

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

   8. Taylor expanded in t around 0 54.3%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{-114} \lor \neg \left(a \leq -1.05 \cdot 10^{-175}\right):\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \log t + \log \left(y \cdot z\right)\\ \end{array} \]

Alternative 13: 59.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{+35} \lor \neg \left(t \leq 8.2 \cdot 10^{+60}\right) \land t \leq 8.5 \cdot 10^{+114}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t 1.05e+35) (and (not (<= t 8.2e+60)) (<= t 8.5e+114)))
   (* a (log t))
   (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= 1.05e+35) || (!(t <= 8.2e+60) && (t <= 8.5e+114))) {
		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 <= 1.05d+35) .or. (.not. (t <= 8.2d+60)) .and. (t <= 8.5d+114)) 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 <= 1.05e+35) || (!(t <= 8.2e+60) && (t <= 8.5e+114))) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= 1.05e+35) or (not (t <= 8.2e+60) and (t <= 8.5e+114)):
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= 1.05e+35) || (!(t <= 8.2e+60) && (t <= 8.5e+114)))
		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 <= 1.05e+35) || (~((t <= 8.2e+60)) && (t <= 8.5e+114)))
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, 1.05e+35], And[N[Not[LessEqual[t, 8.2e+60]], $MachinePrecision], LessEqual[t, 8.5e+114]]], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 1.0499999999999999e35 or 8.2e60 < t < 8.5000000000000001e114

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-+l+ 99.4%

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

      4. +-commutative 99.4%

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

      5. fma-def 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 72.8%

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

   5. Taylor expanded in t around 0 66.9%

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

   6. Taylor expanded in a around inf 53.1%

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

   if 1.0499999999999999e35 < t < 8.2e60 or 8.5000000000000001e114 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 70.3%

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

   5. Taylor expanded in t around inf 88.7%

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

      1. neg-mul-1 88.7%

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

   7. Simplified 88.7%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{+35} \lor \neg \left(t \leq 8.2 \cdot 10^{+60}\right) \land t \leq 8.5 \cdot 10^{+114}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 14: 57.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.6 \cdot 10^{+61} \lor \neg \left(a \leq 1.5 \cdot 10^{+16}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -3.6e+61) (not (<= a 1.5e+16))) (* a (log t)) (- (log y) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3.6e+61) || !(a <= 1.5e+16)) {
		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 <= (-3.6d+61)) .or. (.not. (a <= 1.5d+16))) 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 <= -3.6e+61) || !(a <= 1.5e+16)) {
		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 <= -3.6e+61) or not (a <= 1.5e+16):
		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 <= -3.6e+61) || !(a <= 1.5e+16))
		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 <= -3.6e+61) || ~((a <= 1.5e+16)))
		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, -3.6e+61], N[Not[LessEqual[a, 1.5e+16]], $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 < -3.6000000000000001e61 or 1.5e16 < a

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 73.1%

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

   5. Taylor expanded in t around 0 60.7%

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

   6. Taylor expanded in a around inf 79.8%

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

   if -3.6000000000000001e61 < a < 1.5e16

   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+ 99.6%

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

      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 t around inf 60.3%

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

      1. neg-mul-1 60.3%

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

   6. Simplified 60.3%

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

   7. Taylor expanded in x around 0 47.2%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.6 \cdot 10^{+61} \lor \neg \left(a \leq 1.5 \cdot 10^{+16}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log y - t\\ \end{array} \]

Alternative 15: 41.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 980:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (if (<= t 980.0) (log (+ x y)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 980.0) {
		tmp = log((x + y));
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 980.0d0) then
        tmp = log((x + y))
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 980.0) {
		tmp = Math.log((x + y));
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 980.0:
		tmp = math.log((x + y))
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 980.0)
		tmp = log(Float64(x + y));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 980.0)
		tmp = log((x + y));
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 980.0], N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 980

   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+ 99.4%

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

      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 t around inf 9.6%

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

      1. neg-mul-1 9.6%

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

   6. Simplified 9.6%

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

   7. Taylor expanded in t around 0 9.7%

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

   if 980 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 73.3%

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

   5. Taylor expanded in t around inf 73.8%

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

      1. neg-mul-1 73.8%

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

   7. Simplified 73.8%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 980:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 16: 74.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ a \cdot \log t - t \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
2. 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 99.7%

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

   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 71.8%

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

   1. associate--l+ 71.8%

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

   2. remove-double-neg 71.8%

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

   3. log-rec 71.8%

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

   4. mul-1-neg 71.8%

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

   5. associate--l+ 71.8%

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

   6. fma-def 71.8%

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

   7. sub-neg 71.8%

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

   8. metadata-eval 71.8%

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

   9. +-commutative 71.8%

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

   10. mul-1-neg 71.8%

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

   11. log-rec 71.8%

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

   12. remove-double-neg 71.8%

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

6. Simplified 71.8%

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

7. Taylor expanded in a around inf 75.4%

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

   1. *-commutative 75.4%

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

9. Simplified 75.4%

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

10. Final simplification 75.4%

    \[\leadsto a \cdot \log t - t \]

Alternative 17: 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. +-commutative 99.6%

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

   3. associate-+l+ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-def 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 71.8%

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

5. Taylor expanded in t around inf 40.2%

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

   1. neg-mul-1 40.2%

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

7. Simplified 40.2%

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

8. Final simplification 40.2%

   \[\leadsto -t \]

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 2023257 
(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))))