Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 18.3s

Alternatives: 22

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate--l+ 99.5%

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

   2. +-commutative 99.5%

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

   3. associate-+l+ 99.5%

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

   4. +-commutative 99.5%

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

   5. fma-define 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. Add Preprocessing
5. Add Preprocessing

Alternative 2: 84.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log z + \log \left(x + y\right)\\ t_2 := \log y + \left(\log \left(z \cdot {t}^{-0.5}\right) - t\right)\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 710:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \left(\log \left(z \cdot \left(x + y\right)\right) - t\right)\\ \mathbf{elif}\;t\_1 \leq 858:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log z) (log (+ x y))))
        (t_2 (+ (log y) (- (log (* z (pow t -0.5))) t))))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 710.0)
       (+ (* (log t) (- a 0.5)) (- (log (* z (+ x y))) t))
       (if (<= t_1 858.0) t_2 (* a (log t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(z) + log((x + y));
	double t_2 = log(y) + (log((z * pow(t, -0.5))) - t);
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 710.0) {
		tmp = (log(t) * (a - 0.5)) + (log((z * (x + y))) - t);
	} else if (t_1 <= 858.0) {
		tmp = t_2;
	} else {
		tmp = 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log(z) + log((x + y))
    t_2 = log(y) + (log((z * (t ** (-0.5d0)))) - t)
    if (t_1 <= (-750.0d0)) then
        tmp = t_2
    else if (t_1 <= 710.0d0) then
        tmp = (log(t) * (a - 0.5d0)) + (log((z * (x + y))) - t)
    else if (t_1 <= 858.0d0) then
        tmp = t_2
    else
        tmp = 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 t_1 = Math.log(z) + Math.log((x + y));
	double t_2 = Math.log(y) + (Math.log((z * Math.pow(t, -0.5))) - t);
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 710.0) {
		tmp = (Math.log(t) * (a - 0.5)) + (Math.log((z * (x + y))) - t);
	} else if (t_1 <= 858.0) {
		tmp = t_2;
	} else {
		tmp = a * Math.log(t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(z) + math.log((x + y))
	t_2 = math.log(y) + (math.log((z * math.pow(t, -0.5))) - t)
	tmp = 0
	if t_1 <= -750.0:
		tmp = t_2
	elif t_1 <= 710.0:
		tmp = (math.log(t) * (a - 0.5)) + (math.log((z * (x + y))) - t)
	elif t_1 <= 858.0:
		tmp = t_2
	else:
		tmp = a * math.log(t)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(z) + log(Float64(x + y)))
	t_2 = Float64(log(y) + Float64(log(Float64(z * (t ^ -0.5))) - t))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 710.0)
		tmp = Float64(Float64(log(t) * Float64(a - 0.5)) + Float64(log(Float64(z * Float64(x + y))) - t));
	elseif (t_1 <= 858.0)
		tmp = t_2;
	else
		tmp = Float64(a * log(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(z) + log((x + y));
	t_2 = log(y) + (log((z * (t ^ -0.5))) - t);
	tmp = 0.0;
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 710.0)
		tmp = (log(t) * (a - 0.5)) + (log((z * (x + y))) - t);
	elseif (t_1 <= 858.0)
		tmp = t_2;
	else
		tmp = a * log(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[z], $MachinePrecision] + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[y], $MachinePrecision] + N[(N[Log[N[(z * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 710.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Log[N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 858.0], t$95$2, N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 710 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 858

   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-define 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. Add Preprocessing

   5. Taylor expanded in x around 0 66.7%

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

      1. associate--l+ 66.7%

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

      2. sub-neg 66.7%

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

      3. metadata-eval 66.7%

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

   7. Simplified 66.7%

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

      1. +-commutative 66.7%

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

      2. *-un-lft-identity 66.7%

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

      3. fma-define 66.7%

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

      4. add-log-exp 34.2%

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

      5. sum-log 34.2%

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

      6. exp-to-pow 34.2%

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

   9. Applied egg-rr 34.2%

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

       1. fma-undefine 34.2%

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

       2. *-lft-identity 34.2%

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

   11. Simplified 34.2%

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

   12. Taylor expanded in a around 0 34.1%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 710

   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. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 99.5%

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

      2. +-commutative 99.5%

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

      3. sum-log 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. *-lft-identity 99.6%

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

   6. Simplified 99.6%

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

   if 858 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. +-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+ 99.6%

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

      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-define 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. Add Preprocessing

   5. Taylor expanded in a around inf 55.4%

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

      1. *-commutative 55.4%

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

   7. Simplified 55.4%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(x + y\right) \leq -750:\\ \;\;\;\;\log y + \left(\log \left(z \cdot {t}^{-0.5}\right) - t\right)\\ \mathbf{elif}\;\log z + \log \left(x + y\right) \leq 710:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \left(\log \left(z \cdot \left(x + y\right)\right) - t\right)\\ \mathbf{elif}\;\log z + \log \left(x + y\right) \leq 858:\\ \;\;\;\;\log y + \left(\log \left(z \cdot {t}^{-0.5}\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot \left(a - 0.5\right)\\ t_2 := \log z + \log \left(x + y\right)\\ \mathbf{if}\;t\_2 \leq -750 \lor \neg \left(t\_2 \leq 710\right):\\ \;\;\;\;\log y + \left(\log z + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(\log \left(z \cdot \left(x + y\right)\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) (- a 0.5))) (t_2 (+ (log z) (log (+ x y)))))
   (if (or (<= t_2 -750.0) (not (<= t_2 710.0)))
     (+ (log y) (+ (log z) t_1))
     (+ t_1 (- (log (* z (+ x y))) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * (a - 0.5);
	double t_2 = log(z) + log((x + y));
	double tmp;
	if ((t_2 <= -750.0) || !(t_2 <= 710.0)) {
		tmp = log(y) + (log(z) + t_1);
	} else {
		tmp = t_1 + (log((z * (x + y))) - t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log(t) * (a - 0.5d0)
    t_2 = log(z) + log((x + y))
    if ((t_2 <= (-750.0d0)) .or. (.not. (t_2 <= 710.0d0))) then
        tmp = log(y) + (log(z) + t_1)
    else
        tmp = t_1 + (log((z * (x + y))) - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log(t) * (a - 0.5);
	double t_2 = Math.log(z) + Math.log((x + y));
	double tmp;
	if ((t_2 <= -750.0) || !(t_2 <= 710.0)) {
		tmp = Math.log(y) + (Math.log(z) + t_1);
	} else {
		tmp = t_1 + (Math.log((z * (x + y))) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(t) * (a - 0.5)
	t_2 = math.log(z) + math.log((x + y))
	tmp = 0
	if (t_2 <= -750.0) or not (t_2 <= 710.0):
		tmp = math.log(y) + (math.log(z) + t_1)
	else:
		tmp = t_1 + (math.log((z * (x + y))) - t)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * Float64(a - 0.5))
	t_2 = Float64(log(z) + log(Float64(x + y)))
	tmp = 0.0
	if ((t_2 <= -750.0) || !(t_2 <= 710.0))
		tmp = Float64(log(y) + Float64(log(z) + t_1));
	else
		tmp = Float64(t_1 + Float64(log(Float64(z * Float64(x + y))) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(t) * (a - 0.5);
	t_2 = log(z) + log((x + y));
	tmp = 0.0;
	if ((t_2 <= -750.0) || ~((t_2 <= 710.0)))
		tmp = log(y) + (log(z) + t_1);
	else
		tmp = t_1 + (log((z * (x + y))) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[z], $MachinePrecision] + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -750.0], N[Not[LessEqual[t$95$2, 710.0]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[Log[N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 710 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   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-define 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. Add Preprocessing

   5. Taylor expanded in x around 0 72.1%

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

      1. associate--l+ 72.1%

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

      2. sub-neg 72.1%

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

      3. metadata-eval 72.1%

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

   7. Simplified 72.1%

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

   8. Taylor expanded in t around 0 54.2%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 710

   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. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 99.5%

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

      2. +-commutative 99.5%

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

      3. sum-log 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. *-lft-identity 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(x + y\right) \leq -750 \lor \neg \left(\log z + \log \left(x + y\right) \leq 710\right):\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \left(\log \left(z \cdot \left(x + y\right)\right) - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log z + \log \left(x + y\right)\\ \mathbf{if}\;t\_1 \leq -750 \lor \neg \left(t\_1 \leq 710\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \left(\log \left(z \cdot \left(x + y\right)\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log z) (log (+ x y)))))
   (if (or (<= t_1 -750.0) (not (<= t_1 710.0)))
     (* a (log t))
     (+ (* (log t) (- a 0.5)) (- (log (* z (+ x y))) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(z) + log((x + y));
	double tmp;
	if ((t_1 <= -750.0) || !(t_1 <= 710.0)) {
		tmp = a * log(t);
	} else {
		tmp = (log(t) * (a - 0.5)) + (log((z * (x + y))) - t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(z) + log((x + y))
    if ((t_1 <= (-750.0d0)) .or. (.not. (t_1 <= 710.0d0))) then
        tmp = a * log(t)
    else
        tmp = (log(t) * (a - 0.5d0)) + (log((z * (x + y))) - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log(z) + Math.log((x + y));
	double tmp;
	if ((t_1 <= -750.0) || !(t_1 <= 710.0)) {
		tmp = a * Math.log(t);
	} else {
		tmp = (Math.log(t) * (a - 0.5)) + (Math.log((z * (x + y))) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(z) + math.log((x + y))
	tmp = 0
	if (t_1 <= -750.0) or not (t_1 <= 710.0):
		tmp = a * math.log(t)
	else:
		tmp = (math.log(t) * (a - 0.5)) + (math.log((z * (x + y))) - t)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(z) + log(Float64(x + y)))
	tmp = 0.0
	if ((t_1 <= -750.0) || !(t_1 <= 710.0))
		tmp = Float64(a * log(t));
	else
		tmp = Float64(Float64(log(t) * Float64(a - 0.5)) + Float64(log(Float64(z * Float64(x + y))) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(z) + log((x + y));
	tmp = 0.0;
	if ((t_1 <= -750.0) || ~((t_1 <= 710.0)))
		tmp = a * log(t);
	else
		tmp = (log(t) * (a - 0.5)) + (log((z * (x + y))) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[z], $MachinePrecision] + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -750.0], N[Not[LessEqual[t$95$1, 710.0]], $MachinePrecision]], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Log[N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 710 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   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-define 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. Add Preprocessing

   5. Taylor expanded in a around inf 49.3%

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

      1. *-commutative 49.3%

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

   7. Simplified 49.3%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 710

   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. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 99.5%

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

      2. +-commutative 99.5%

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

      3. sum-log 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. *-lft-identity 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(x + y\right) \leq -750 \lor \neg \left(\log z + \log \left(x + y\right) \leq 710\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \left(\log \left(z \cdot \left(x + y\right)\right) - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log z + \log \left(x + y\right)\\ \mathbf{if}\;t\_1 \leq -750 \lor \neg \left(t\_1 \leq 710\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot \left(a - 0.5\right) + \log \left(z \cdot y\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log z) (log (+ x y)))))
   (if (or (<= t_1 -750.0) (not (<= t_1 710.0)))
     (* a (log t))
     (- (+ (* (log t) (- a 0.5)) (log (* z y))) t))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(z) + log((x + y));
	double tmp;
	if ((t_1 <= -750.0) || !(t_1 <= 710.0)) {
		tmp = a * log(t);
	} else {
		tmp = ((log(t) * (a - 0.5)) + log((z * 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) :: t_1
    real(8) :: tmp
    t_1 = log(z) + log((x + y))
    if ((t_1 <= (-750.0d0)) .or. (.not. (t_1 <= 710.0d0))) then
        tmp = a * log(t)
    else
        tmp = ((log(t) * (a - 0.5d0)) + log((z * y))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log(z) + Math.log((x + y));
	double tmp;
	if ((t_1 <= -750.0) || !(t_1 <= 710.0)) {
		tmp = a * Math.log(t);
	} else {
		tmp = ((Math.log(t) * (a - 0.5)) + Math.log((z * y))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(z) + math.log((x + y))
	tmp = 0
	if (t_1 <= -750.0) or not (t_1 <= 710.0):
		tmp = a * math.log(t)
	else:
		tmp = ((math.log(t) * (a - 0.5)) + math.log((z * y))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(z) + log(Float64(x + y)))
	tmp = 0.0
	if ((t_1 <= -750.0) || !(t_1 <= 710.0))
		tmp = Float64(a * log(t));
	else
		tmp = Float64(Float64(Float64(log(t) * Float64(a - 0.5)) + log(Float64(z * y))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(z) + log((x + y));
	tmp = 0.0;
	if ((t_1 <= -750.0) || ~((t_1 <= 710.0)))
		tmp = a * log(t);
	else
		tmp = ((log(t) * (a - 0.5)) + log((z * y))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[z], $MachinePrecision] + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -750.0], N[Not[LessEqual[t$95$1, 710.0]], $MachinePrecision]], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 710 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   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-define 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. Add Preprocessing

   5. Taylor expanded in a around inf 49.3%

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

      1. *-commutative 49.3%

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

   7. Simplified 49.3%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 710

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-define 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. associate--l+ 99.5%

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

   3. Simplified 99.5%

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

      1. fma-undefine 99.5%

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

      2. +-commutative 99.5%

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

      3. add-sqr-sqrt 49.5%

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

      4. pow2 49.5%

         \[\leadsto \color{blue}{{\left(\sqrt{\left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \log t}\right)}^{2}} \]

      5. +-commutative 49.5%

         \[\leadsto {\left(\sqrt{\color{blue}{\left(a + -0.5\right) \cdot \log t + \left(\log \left(x + y\right) + \left(\log z - t\right)\right)}}\right)}^{2} \]

      6. fma-undefine 49.5%

         \[\leadsto {\left(\sqrt{\color{blue}{\mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right) + \left(\log z - t\right)\right)}}\right)}^{2} \]

      7. associate-+r- 49.5%

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(a + -0.5, \log t, \color{blue}{\left(\log \left(x + y\right) + \log z\right) - t}\right)}\right)}^{2} \]

      8. +-commutative 49.5%

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(a + -0.5, \log t, \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - t\right)}\right)}^{2} \]

      9. sum-log 49.6%

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(a + -0.5, \log t, \color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - t\right)}\right)}^{2} \]

   6. Applied egg-rr 49.6%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(a + -0.5, \log t, \log \left(z \cdot \left(x + y\right)\right) - t\right)}\right)}^{2}} \]

   7. Taylor expanded in x around 0 62.7%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(x + y\right) \leq -750 \lor \neg \left(\log z + \log \left(x + y\right) \leq 710\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot \left(a - 0.5\right) + \log \left(z \cdot y\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-+l- 99.5%

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

   2. associate--l+ 99.5%

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

   3. sub-neg 99.5%

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

   4. +-commutative 99.5%

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

   5. *-commutative 99.5%

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

   6. distribute-rgt-neg-in 99.5%

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

   7. fma-undefine 99.5%

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

   8. sub-neg 99.5%

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

   9. +-commutative 99.5%

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

   10. distribute-neg-in 99.5%

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

   11. metadata-eval 99.5%

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

   12. metadata-eval 99.5%

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

   13. unsub-neg 99.5%

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

3. Simplified 99.5%

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

Alternative 7: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.038:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log y + \left(\left(\log z + a \cdot \log t\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 0.038)
   (+ (log y) (+ (log z) (* (log t) (- a 0.5))))
   (+ (log y) (- (+ (log z) (* a (log t))) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.038) {
		tmp = log(y) + (log(z) + (log(t) * (a - 0.5)));
	} else {
		tmp = log(y) + ((log(z) + (a * log(t))) - t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 0.038d0) then
        tmp = log(y) + (log(z) + (log(t) * (a - 0.5d0)))
    else
        tmp = log(y) + ((log(z) + (a * log(t))) - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.038) {
		tmp = Math.log(y) + (Math.log(z) + (Math.log(t) * (a - 0.5)));
	} else {
		tmp = Math.log(y) + ((Math.log(z) + (a * Math.log(t))) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 0.038:
		tmp = math.log(y) + (math.log(z) + (math.log(t) * (a - 0.5)))
	else:
		tmp = math.log(y) + ((math.log(z) + (a * math.log(t))) - t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 0.038)
		tmp = Float64(log(y) + Float64(log(z) + Float64(log(t) * Float64(a - 0.5))));
	else
		tmp = Float64(log(y) + Float64(Float64(log(z) + Float64(a * log(t))) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 0.038)
		tmp = log(y) + (log(z) + (log(t) * (a - 0.5)));
	else
		tmp = log(y) + ((log(z) + (a * log(t))) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 0.038], N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] + N[(N[(N[Log[z], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 0.0379999999999999991

   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-define 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. Add Preprocessing

   5. Taylor expanded in x around 0 65.2%

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

      1. associate--l+ 65.2%

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

      2. sub-neg 65.2%

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

      3. metadata-eval 65.2%

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

   7. Simplified 65.2%

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

   8. Taylor expanded in t around 0 64.8%

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

   if 0.0379999999999999991 < t

   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-define 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. Add Preprocessing

   5. Taylor expanded in x around 0 68.8%

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

      1. associate--l+ 68.8%

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

      2. sub-neg 68.8%

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

      3. metadata-eval 68.8%

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

   7. Simplified 68.8%

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

   8. Taylor expanded in a around inf 68.8%

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

      1. *-commutative 68.8%

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

   10. Simplified 68.8%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.038:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log y + \left(\left(\log z + a \cdot \log t\right) - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate--l+ 99.5%

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

   2. sub-neg 99.5%

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

   3. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 9: 68.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return (log(z) - t) + (((a + -0.5) * log(t)) + 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) + (((a + (-0.5d0)) * log(t)) + log(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate--l+ 99.5%

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

   2. +-commutative 99.5%

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

   3. associate-+l+ 99.5%

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

   4. +-commutative 99.5%

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

   5. fma-define 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. Add Preprocessing

5. Taylor expanded in x around 0 66.8%

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

   1. +-commutative 66.8%

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

   2. sub-neg 66.8%

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

   3. metadata-eval 66.8%

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

7. Simplified 66.8%

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

8. Final simplification 66.8%

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

Alternative 10: 68.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return log(y) + ((log(z) + ((a + -0.5) * 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 = log(y) + ((log(z) + ((a + (-0.5d0)) * log(t))) - t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate--l+ 99.5%

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

   2. +-commutative 99.5%

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

   3. associate-+l+ 99.5%

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

   4. +-commutative 99.5%

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

   5. fma-define 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. Add Preprocessing

5. Taylor expanded in x around 0 66.8%

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

   1. associate--l+ 66.8%

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

   2. sub-neg 66.8%

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

   3. metadata-eval 66.8%

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

7. Simplified 66.8%

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

8. Final simplification 66.8%

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

Alternative 11: 63.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -2600:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 7.8 \cdot 10^{-67}:\\ \;\;\;\;\left(\log z - t\right) + \log \left(x + y\right)\\ \mathbf{elif}\;a \leq 2.8:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{\left(a + -0.5\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= a -2600.0)
     t_1
     (if (<= a 7.8e-67)
       (+ (- (log z) t) (log (+ x y)))
       (if (<= a 2.8) (log (* y (* z (pow t (+ a -0.5))))) 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 <= -2600.0) {
		tmp = t_1;
	} else if (a <= 7.8e-67) {
		tmp = (log(z) - t) + log((x + y));
	} else if (a <= 2.8) {
		tmp = log((y * (z * pow(t, (a + -0.5)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if (a <= (-2600.0d0)) then
        tmp = t_1
    else if (a <= 7.8d-67) then
        tmp = (log(z) - t) + log((x + y))
    else if (a <= 2.8d0) then
        tmp = log((y * (z * (t ** (a + (-0.5d0))))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if (a <= -2600.0) {
		tmp = t_1;
	} else if (a <= 7.8e-67) {
		tmp = (Math.log(z) - t) + Math.log((x + y));
	} else if (a <= 2.8) {
		tmp = Math.log((y * (z * Math.pow(t, (a + -0.5)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if a <= -2600.0:
		tmp = t_1
	elif a <= 7.8e-67:
		tmp = (math.log(z) - t) + math.log((x + y))
	elif a <= 2.8:
		tmp = math.log((y * (z * math.pow(t, (a + -0.5)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (a <= -2600.0)
		tmp = t_1;
	elseif (a <= 7.8e-67)
		tmp = Float64(Float64(log(z) - t) + log(Float64(x + y)));
	elseif (a <= 2.8)
		tmp = log(Float64(y * Float64(z * (t ^ Float64(a + -0.5)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if (a <= -2600.0)
		tmp = t_1;
	elseif (a <= 7.8e-67)
		tmp = (log(z) - t) + log((x + y));
	elseif (a <= 2.8)
		tmp = log((y * (z * (t ^ (a + -0.5)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2600.0], t$95$1, If[LessEqual[a, 7.8e-67], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.8], N[Log[N[(y * N[(z * N[Power[t, N[(a + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2600 or 2.7999999999999998 < 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+ 99.6%

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

      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-define 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. Add Preprocessing

   5. Taylor expanded in a around inf 77.0%

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

      1. *-commutative 77.0%

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

   7. Simplified 77.0%

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

   if -2600 < a < 7.7999999999999997e-67

   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%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-undefine 99.5%

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

      8. sub-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. distribute-neg-in 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. unsub-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in t around inf 51.4%

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

   if 7.7999999999999997e-67 < a < 2.7999999999999998

   1. Initial program 98.7%

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

      1. associate--l+ 98.7%

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

      2. +-commutative 98.7%

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

      3. associate-+l+ 99.1%

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

      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-define 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. Add Preprocessing

   5. Taylor expanded in x around 0 54.4%

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

      1. associate--l+ 54.4%

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

      2. sub-neg 54.4%

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

      3. metadata-eval 54.4%

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

   7. Simplified 54.4%

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

      1. +-commutative 54.4%

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

      2. *-un-lft-identity 54.4%

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

      3. fma-define 54.4%

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

      4. add-log-exp 54.4%

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

      5. sum-log 45.8%

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

      6. exp-to-pow 45.8%

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

   9. Applied egg-rr 45.8%

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

       1. fma-undefine 45.8%

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

       2. *-lft-identity 45.8%

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

   11. Simplified 45.8%

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

   12. Taylor expanded in t around 0 36.8%

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

       1. exp-to-pow 36.8%

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

       2. sub-neg 36.8%

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

       3. metadata-eval 36.8%

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

   14. Simplified 36.8%

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

       1. sum-log 37.4%

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

   16. Applied egg-rr 37.4%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2600:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a \leq 7.8 \cdot 10^{-67}:\\ \;\;\;\;\left(\log z - t\right) + \log \left(x + y\right)\\ \mathbf{elif}\;a \leq 2.8:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{\left(a + -0.5\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -2000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-66}:\\ \;\;\;\;\left(\log z - t\right) + \log \left(x + y\right)\\ \mathbf{elif}\;a \leq 3.8:\\ \;\;\;\;\log \left(\left(z \cdot y\right) \cdot {t}^{\left(a + -0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= a -2000.0)
     t_1
     (if (<= a 1.3e-66)
       (+ (- (log z) t) (log (+ x y)))
       (if (<= a 3.8) (log (* (* z y) (pow t (+ a -0.5)))) 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 <= -2000.0) {
		tmp = t_1;
	} else if (a <= 1.3e-66) {
		tmp = (log(z) - t) + log((x + y));
	} else if (a <= 3.8) {
		tmp = log(((z * y) * pow(t, (a + -0.5))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if (a <= (-2000.0d0)) then
        tmp = t_1
    else if (a <= 1.3d-66) then
        tmp = (log(z) - t) + log((x + y))
    else if (a <= 3.8d0) then
        tmp = log(((z * y) * (t ** (a + (-0.5d0)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if (a <= -2000.0) {
		tmp = t_1;
	} else if (a <= 1.3e-66) {
		tmp = (Math.log(z) - t) + Math.log((x + y));
	} else if (a <= 3.8) {
		tmp = Math.log(((z * y) * Math.pow(t, (a + -0.5))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if a <= -2000.0:
		tmp = t_1
	elif a <= 1.3e-66:
		tmp = (math.log(z) - t) + math.log((x + y))
	elif a <= 3.8:
		tmp = math.log(((z * y) * math.pow(t, (a + -0.5))))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (a <= -2000.0)
		tmp = t_1;
	elseif (a <= 1.3e-66)
		tmp = Float64(Float64(log(z) - t) + log(Float64(x + y)));
	elseif (a <= 3.8)
		tmp = log(Float64(Float64(z * y) * (t ^ Float64(a + -0.5))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if (a <= -2000.0)
		tmp = t_1;
	elseif (a <= 1.3e-66)
		tmp = (log(z) - t) + log((x + y));
	elseif (a <= 3.8)
		tmp = log(((z * y) * (t ^ (a + -0.5))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2000.0], t$95$1, If[LessEqual[a, 1.3e-66], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.8], N[Log[N[(N[(z * y), $MachinePrecision] * N[Power[t, N[(a + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2e3 or 3.7999999999999998 < 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+ 99.6%

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

      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-define 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. Add Preprocessing

   5. Taylor expanded in a around inf 77.0%

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

      1. *-commutative 77.0%

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

   7. Simplified 77.0%

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

   if -2e3 < a < 1.2999999999999999e-66

   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%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-undefine 99.5%

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

      8. sub-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. distribute-neg-in 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. unsub-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in t around inf 51.4%

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

   if 1.2999999999999999e-66 < a < 3.7999999999999998

   1. Initial program 98.7%

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

      1. associate--l+ 98.7%

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

      2. +-commutative 98.7%

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

      3. associate-+l+ 99.1%

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

      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-define 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. Add Preprocessing

   5. Taylor expanded in x around 0 54.4%

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

      1. associate--l+ 54.4%

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

      2. sub-neg 54.4%

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

      3. metadata-eval 54.4%

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

   7. Simplified 54.4%

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

      1. +-commutative 54.4%

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

      2. *-un-lft-identity 54.4%

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

      3. fma-define 54.4%

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

      4. add-log-exp 54.4%

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

      5. sum-log 45.8%

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

      6. exp-to-pow 45.8%

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

   9. Applied egg-rr 45.8%

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

       1. fma-undefine 45.8%

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

       2. *-lft-identity 45.8%

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

   11. Simplified 45.8%

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

   12. Taylor expanded in t around 0 36.8%

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

       1. exp-to-pow 36.8%

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

       2. sub-neg 36.8%

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

       3. metadata-eval 36.8%

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

   14. Simplified 36.8%

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

       1. *-un-lft-identity 36.8%

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

       2. sum-log 37.4%

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

   16. Applied egg-rr 37.4%

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

       1. *-lft-identity 37.4%

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

       2. *-commutative 37.4%

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

       3. associate-*r* 37.2%

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

       4. +-commutative 37.2%

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

   18. Simplified 37.2%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2000:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-66}:\\ \;\;\;\;\left(\log z - t\right) + \log \left(x + y\right)\\ \mathbf{elif}\;a \leq 3.8:\\ \;\;\;\;\log \left(\left(z \cdot y\right) \cdot {t}^{\left(a + -0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{-10} \lor \neg \left(a \leq 8.5\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(z \cdot y\right) \cdot {t}^{\left(a + -0.5\right)}\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -3e-10) (not (<= a 8.5)))
   (* a (log t))
   (- (log (* (* z y) (pow t (+ a -0.5)))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3e-10) || !(a <= 8.5)) {
		tmp = a * log(t);
	} else {
		tmp = log(((z * y) * pow(t, (a + -0.5)))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-3d-10)) .or. (.not. (a <= 8.5d0))) then
        tmp = a * log(t)
    else
        tmp = log(((z * y) * (t ** (a + (-0.5d0))))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3e-10) || !(a <= 8.5)) {
		tmp = a * Math.log(t);
	} else {
		tmp = Math.log(((z * y) * Math.pow(t, (a + -0.5)))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -3e-10) or not (a <= 8.5):
		tmp = a * math.log(t)
	else:
		tmp = math.log(((z * y) * math.pow(t, (a + -0.5)))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -3e-10) || !(a <= 8.5))
		tmp = Float64(a * log(t));
	else
		tmp = Float64(log(Float64(Float64(z * y) * (t ^ Float64(a + -0.5)))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -3e-10) || ~((a <= 8.5)))
		tmp = a * log(t);
	else
		tmp = log(((z * y) * (t ^ (a + -0.5)))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -3e-10], N[Not[LessEqual[a, 8.5]], $MachinePrecision]], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(N[(z * y), $MachinePrecision] * N[Power[t, N[(a + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3e-10 or 8.5 < a

   1. Initial program 99.7%

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

      1. 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+ 99.6%

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

      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-define 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. Add Preprocessing

   5. Taylor expanded in a around inf 75.9%

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

      1. *-commutative 75.9%

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

   7. Simplified 75.9%

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

   if -3e-10 < a < 8.5

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      2. +-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-define 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. Add Preprocessing

   5. Taylor expanded in x around 0 59.9%

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

      1. associate--l+ 59.9%

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

      2. sub-neg 59.9%

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

      3. metadata-eval 59.9%

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

   7. Simplified 59.9%

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

      1. associate-+r- 59.9%

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

      2. add-log-exp 53.3%

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

      3. sum-log 40.9%

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

      4. exp-sum 41.0%

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

      5. add-exp-log 41.0%

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

      6. exp-to-pow 41.1%

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

   9. Applied egg-rr 41.1%

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

       1. associate-*r* 41.8%

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

   11. Simplified 41.8%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{-10} \lor \neg \left(a \leq 8.5\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(z \cdot y\right) \cdot {t}^{\left(a + -0.5\right)}\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{-10} \lor \neg \left(a \leq 6.2\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{\left(a + -0.5\right)}\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -3e-10) (not (<= a 6.2)))
   (* a (log t))
   (- (log (* y (* z (pow t (+ a -0.5))))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3e-10) || !(a <= 6.2)) {
		tmp = a * log(t);
	} else {
		tmp = log((y * (z * pow(t, (a + -0.5))))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-3d-10)) .or. (.not. (a <= 6.2d0))) then
        tmp = a * log(t)
    else
        tmp = log((y * (z * (t ** (a + (-0.5d0)))))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3e-10) || !(a <= 6.2)) {
		tmp = a * Math.log(t);
	} else {
		tmp = Math.log((y * (z * Math.pow(t, (a + -0.5))))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -3e-10) or not (a <= 6.2):
		tmp = a * math.log(t)
	else:
		tmp = math.log((y * (z * math.pow(t, (a + -0.5))))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -3e-10) || !(a <= 6.2))
		tmp = Float64(a * log(t));
	else
		tmp = Float64(log(Float64(y * Float64(z * (t ^ Float64(a + -0.5))))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -3e-10) || ~((a <= 6.2)))
		tmp = a * log(t);
	else
		tmp = log((y * (z * (t ^ (a + -0.5))))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -3e-10], N[Not[LessEqual[a, 6.2]], $MachinePrecision]], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(y * N[(z * N[Power[t, N[(a + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3e-10 or 6.20000000000000018 < 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+ 99.6%

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

      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-define 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. Add Preprocessing

   5. Taylor expanded in a around inf 75.9%

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

      1. *-commutative 75.9%

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

   7. Simplified 75.9%

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

   if -3e-10 < a < 6.20000000000000018

   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-define 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. Add Preprocessing

   5. Taylor expanded in x around 0 59.9%

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

      1. associate--l+ 59.9%

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

      2. sub-neg 59.9%

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

      3. metadata-eval 59.9%

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

   7. Simplified 59.9%

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

      1. associate-+r- 59.9%

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

      2. add-log-exp 53.3%

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

      3. sum-log 40.9%

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

      4. exp-sum 41.0%

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

      5. add-exp-log 41.0%

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

      6. exp-to-pow 41.1%

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

   9. Applied egg-rr 41.1%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{-10} \lor \neg \left(a \leq 6.2\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{\left(a + -0.5\right)}\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 66.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2600 \lor \neg \left(a \leq 850000000\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \log \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -2600.0) (not (<= a 850000000.0)))
   (* a (log t))
   (+ (- (log z) t) (log (+ x y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2600.0) || !(a <= 850000000.0)) {
		tmp = a * log(t);
	} else {
		tmp = (log(z) - t) + log((x + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-2600.0d0)) .or. (.not. (a <= 850000000.0d0))) then
        tmp = a * log(t)
    else
        tmp = (log(z) - t) + log((x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2600.0) || !(a <= 850000000.0)) {
		tmp = a * Math.log(t);
	} else {
		tmp = (Math.log(z) - t) + Math.log((x + y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -2600.0) or not (a <= 850000000.0):
		tmp = a * math.log(t)
	else:
		tmp = (math.log(z) - t) + math.log((x + y))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -2600.0) || !(a <= 850000000.0))
		tmp = Float64(a * log(t));
	else
		tmp = Float64(Float64(log(z) - t) + log(Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -2600.0) || ~((a <= 850000000.0)))
		tmp = a * log(t);
	else
		tmp = (log(z) - t) + log((x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -2600.0], N[Not[LessEqual[a, 850000000.0]], $MachinePrecision]], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2600 or 8.5e8 < 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+ 99.6%

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

      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-define 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. Add Preprocessing

   5. Taylor expanded in a around inf 77.0%

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

      1. *-commutative 77.0%

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

   7. Simplified 77.0%

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

   if -2600 < a < 8.5e8

   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%

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

      2. associate--l+ 99.4%

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

      3. sub-neg 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. fma-undefine 99.4%

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

      8. sub-neg 99.4%

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

      9. +-commutative 99.4%

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

      10. distribute-neg-in 99.4%

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

      11. metadata-eval 99.4%

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

      12. metadata-eval 99.4%

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

      13. unsub-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in t around inf 49.8%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2600 \lor \neg \left(a \leq 850000000\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \log \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 58.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2600 \lor \neg \left(a \leq 16500000\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log z + \left(\log y - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -2600.0) (not (<= a 16500000.0)))
   (* a (log t))
   (+ (log z) (- (log y) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2600.0) || !(a <= 16500000.0)) {
		tmp = a * log(t);
	} else {
		tmp = log(z) + (log(y) - t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-2600.0d0)) .or. (.not. (a <= 16500000.0d0))) then
        tmp = a * log(t)
    else
        tmp = log(z) + (log(y) - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2600.0) || !(a <= 16500000.0)) {
		tmp = a * Math.log(t);
	} else {
		tmp = Math.log(z) + (Math.log(y) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -2600.0) or not (a <= 16500000.0):
		tmp = a * math.log(t)
	else:
		tmp = math.log(z) + (math.log(y) - t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -2600.0) || !(a <= 16500000.0))
		tmp = Float64(a * log(t));
	else
		tmp = Float64(log(z) + Float64(log(y) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -2600.0) || ~((a <= 16500000.0)))
		tmp = a * log(t);
	else
		tmp = log(z) + (log(y) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -2600.0], N[Not[LessEqual[a, 16500000.0]], $MachinePrecision]], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], N[(N[Log[z], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2600 or 1.65e7 < 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+ 99.6%

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

      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-define 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. Add Preprocessing

   5. Taylor expanded in a around inf 77.0%

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

      1. *-commutative 77.0%

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

   7. Simplified 77.0%

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

   if -2600 < a < 1.65e7

   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%

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

      2. associate--l+ 99.4%

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

      3. sub-neg 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. fma-undefine 99.4%

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

      8. sub-neg 99.4%

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

      9. +-commutative 99.4%

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

      10. distribute-neg-in 99.4%

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

      11. metadata-eval 99.4%

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

      12. metadata-eval 99.4%

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

      13. unsub-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in t around inf 49.8%

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

   6. Taylor expanded in x around 0 36.6%

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

      1. remove-double-neg 36.6%

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

      2. log-rec 36.6%

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

      3. mul-1-neg 36.6%

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

      4. +-commutative 36.6%

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

      5. associate--l+ 36.6%

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

      6. mul-1-neg 36.6%

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

      7. log-rec 36.6%

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

      8. remove-double-neg 36.6%

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

   8. Simplified 36.6%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2600 \lor \neg \left(a \leq 16500000\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log z + \left(\log y - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 57.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2600 \lor \neg \left(a \leq 550000000000\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -2600.0) (not (<= a 550000000000.0)))
   (* a (log t))
   (- (log y) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2600.0) || !(a <= 550000000000.0)) {
		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 <= (-2600.0d0)) .or. (.not. (a <= 550000000000.0d0))) 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 <= -2600.0) || !(a <= 550000000000.0)) {
		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 <= -2600.0) or not (a <= 550000000000.0):
		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 <= -2600.0) || !(a <= 550000000000.0))
		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 <= -2600.0) || ~((a <= 550000000000.0)))
		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, -2600.0], N[Not[LessEqual[a, 550000000000.0]], $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 < -2600 or 5.5e11 < 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+ 99.6%

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

      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-define 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. Add Preprocessing

   5. Taylor expanded in a around inf 77.0%

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

      1. *-commutative 77.0%

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

   7. Simplified 77.0%

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

   if -2600 < a < 5.5e11

   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-define 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. Add Preprocessing

   5. Taylor expanded in x around 0 59.7%

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

      1. associate--l+ 59.7%

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

      2. sub-neg 59.7%

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

      3. metadata-eval 59.7%

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

   7. Simplified 59.7%

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

      1. add-sqr-sqrt 0.0%

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

      2. pow2 0.0%

         \[\leadsto \log y + \left(\left(\log z + \log t \cdot \color{blue}{{\left(\sqrt{a + -0.5}\right)}^{2}}\right) - t\right) \]

   9. Applied egg-rr 0.0%

      \[\leadsto \log y + \left(\left(\log z + \log t \cdot \color{blue}{{\left(\sqrt{a + -0.5}\right)}^{2}}\right) - t\right) \]

   10. Taylor expanded in t around inf 59.8%

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

       1. associate--l+ 59.8%

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

       2. mul-1-neg 59.8%

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

       3. sub-neg 59.8%

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

       4. metadata-eval 59.8%

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

       5. associate-/l* 59.7%

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

       6. distribute-lft-neg-in 59.7%

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

       7. log-rec 59.7%

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

       8. remove-double-neg 59.7%

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

       9. sub-neg 59.7%

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

       10. metadata-eval 59.7%

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

   12. Simplified 59.7%

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

   13. Taylor expanded in t around inf 35.8%

       \[\leadsto \log y + t \cdot \color{blue}{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2600 \lor \neg \left(a \leq 550000000000\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 62.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2600 \lor \neg \left(a \leq 1200\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-1 + t \cdot \left(-1 + \frac{1}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -2600.0) (not (<= a 1200.0)))
   (* a (log t))
   (+ -1.0 (* t (+ -1.0 (/ 1.0 t))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -2600.0) || !(a <= 1200.0)) {
		tmp = a * log(t);
	} else {
		tmp = -1.0 + (t * (-1.0 + (1.0 / 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 <= (-2600.0d0)) .or. (.not. (a <= 1200.0d0))) then
        tmp = a * log(t)
    else
        tmp = (-1.0d0) + (t * ((-1.0d0) + (1.0d0 / 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 <= -2600.0) || !(a <= 1200.0)) {
		tmp = a * Math.log(t);
	} else {
		tmp = -1.0 + (t * (-1.0 + (1.0 / t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -2600.0) or not (a <= 1200.0):
		tmp = a * math.log(t)
	else:
		tmp = -1.0 + (t * (-1.0 + (1.0 / t)))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -2600.0) || !(a <= 1200.0))
		tmp = Float64(a * log(t));
	else
		tmp = Float64(-1.0 + Float64(t * Float64(-1.0 + Float64(1.0 / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -2600.0) || ~((a <= 1200.0)))
		tmp = a * log(t);
	else
		tmp = -1.0 + (t * (-1.0 + (1.0 / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -2600.0], N[Not[LessEqual[a, 1200.0]], $MachinePrecision]], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(t * N[(-1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2600 or 1200 < 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+ 99.6%

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

      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-define 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. Add Preprocessing

   5. Taylor expanded in a around inf 77.0%

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

      1. *-commutative 77.0%

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

   7. Simplified 77.0%

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

   if -2600 < a < 1200

   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-define 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. Add Preprocessing

   5. Taylor expanded in t around inf 39.8%

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

      1. neg-mul-1 39.8%

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

   7. Simplified 39.8%

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

      1. expm1-log1p-u 1.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-t\right)\right)} \]

      2. expm1-undefine 1.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]

   9. Applied egg-rr 1.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]
   10. Step-by-step derivation

       1. sub-neg 1.9%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} + \left(-1\right)} \]

       2. log1p-undefine 1.9%

          \[\leadsto e^{\color{blue}{\log \left(1 + \left(-t\right)\right)}} + \left(-1\right) \]

       3. rem-exp-log 40.1%

          \[\leadsto \color{blue}{\left(1 + \left(-t\right)\right)} + \left(-1\right) \]

       4. unsub-neg 40.1%

          \[\leadsto \color{blue}{\left(1 - t\right)} + \left(-1\right) \]

       5. metadata-eval 40.1%

          \[\leadsto \left(1 - t\right) + \color{blue}{-1} \]

   11. Simplified 40.1%

       \[\leadsto \color{blue}{\left(1 - t\right) + -1} \]

   12. Taylor expanded in t around inf 40.1%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2600 \lor \neg \left(a \leq 1200\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-1 + t \cdot \left(-1 + \frac{1}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 37.8% accurate, 34.8× speedup

Math

\[\begin{array}{l} \\ -1 + t \cdot \left(-1 + \frac{1}{t}\right) \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ -1.0 (* t (+ -1.0 (/ 1.0 t)))))

C

double code(double x, double y, double z, double t, double a) {
	return -1.0 + (t * (-1.0 + (1.0 / 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 = (-1.0d0) + (t * ((-1.0d0) + (1.0d0 / t)))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return -1.0 + (t * (-1.0 + (1.0 / t)));
}

Python

def code(x, y, z, t, a):
	return -1.0 + (t * (-1.0 + (1.0 / t)))

Julia

function code(x, y, z, t, a)
	return Float64(-1.0 + Float64(t * Float64(-1.0 + Float64(1.0 / t))))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = -1.0 + (t * (-1.0 + (1.0 / t)));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(-1.0 + N[(t * N[(-1.0 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. associate--l+ 99.5%

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

   2. +-commutative 99.5%

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

   3. associate-+l+ 99.5%

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

   4. +-commutative 99.5%

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

   5. fma-define 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. Add Preprocessing

5. Taylor expanded in t around inf 29.6%

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

   1. neg-mul-1 29.6%

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

7. Simplified 29.6%

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

   1. expm1-log1p-u 1.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-t\right)\right)} \]

   2. expm1-undefine 1.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]

9. Applied egg-rr 1.5%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]
10. Step-by-step derivation

    1. sub-neg 1.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} + \left(-1\right)} \]

    2. log1p-undefine 1.5%

       \[\leadsto e^{\color{blue}{\log \left(1 + \left(-t\right)\right)}} + \left(-1\right) \]

    3. rem-exp-log 29.7%

       \[\leadsto \color{blue}{\left(1 + \left(-t\right)\right)} + \left(-1\right) \]

    4. unsub-neg 29.7%

       \[\leadsto \color{blue}{\left(1 - t\right)} + \left(-1\right) \]

    5. metadata-eval 29.7%

       \[\leadsto \left(1 - t\right) + \color{blue}{-1} \]

11. Simplified 29.7%

    \[\leadsto \color{blue}{\left(1 - t\right) + -1} \]

12. Taylor expanded in t around inf 29.7%

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

13. Final simplification 29.7%

    \[\leadsto -1 + t \cdot \left(-1 + \frac{1}{t}\right) \]
14. Add Preprocessing

Alternative 20: 37.8% accurate, 62.6× speedup

Math

\[\begin{array}{l} \\ -1 + \left(1 - t\right) \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ -1.0 (- 1.0 t)))

C

double code(double x, double y, double z, double t, double a) {
	return -1.0 + (1.0 - 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 = (-1.0d0) + (1.0d0 - t)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return -1.0 + (1.0 - t);
}

Python

def code(x, y, z, t, a):
	return -1.0 + (1.0 - t)

Julia

function code(x, y, z, t, a)
	return Float64(-1.0 + Float64(1.0 - t))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = -1.0 + (1.0 - t);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(-1.0 + N[(1.0 - t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. associate--l+ 99.5%

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

   2. +-commutative 99.5%

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

   3. associate-+l+ 99.5%

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

   4. +-commutative 99.5%

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

   5. fma-define 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. Add Preprocessing

5. Taylor expanded in t around inf 29.6%

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

   1. neg-mul-1 29.6%

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

7. Simplified 29.6%

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

   1. expm1-log1p-u 1.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-t\right)\right)} \]

   2. expm1-undefine 1.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]

9. Applied egg-rr 1.5%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]
10. Step-by-step derivation

    1. sub-neg 1.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} + \left(-1\right)} \]

    2. log1p-undefine 1.5%

       \[\leadsto e^{\color{blue}{\log \left(1 + \left(-t\right)\right)}} + \left(-1\right) \]

    3. rem-exp-log 29.7%

       \[\leadsto \color{blue}{\left(1 + \left(-t\right)\right)} + \left(-1\right) \]

    4. unsub-neg 29.7%

       \[\leadsto \color{blue}{\left(1 - t\right)} + \left(-1\right) \]

    5. metadata-eval 29.7%

       \[\leadsto \left(1 - t\right) + \color{blue}{-1} \]

11. Simplified 29.7%

    \[\leadsto \color{blue}{\left(1 - t\right) + -1} \]

12. Final simplification 29.7%

    \[\leadsto -1 + \left(1 - t\right) \]
13. Add Preprocessing

Alternative 21: 37.8% accurate, 156.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate--l+ 99.5%

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

   2. +-commutative 99.5%

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

   3. associate-+l+ 99.5%

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

   4. +-commutative 99.5%

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

   5. fma-define 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. Add Preprocessing

5. Taylor expanded in t around inf 29.6%

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

   1. neg-mul-1 29.6%

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

7. Simplified 29.6%

   \[\leadsto \color{blue}{-t} \]
8. Add Preprocessing

Alternative 22: 2.4% accurate, 313.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 0.0)

C

double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

Python

def code(x, y, z, t, a):
	return 0.0

Julia

function code(x, y, z, t, a)
	return 0.0
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_, t_, a_] := 0.0

Derivation

1. Initial program 99.5%

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

   1. associate--l+ 99.5%

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

   2. +-commutative 99.5%

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

   3. associate-+l+ 99.5%

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

   4. +-commutative 99.5%

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

   5. fma-define 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. Add Preprocessing

5. Taylor expanded in t around inf 29.6%

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

   1. neg-mul-1 29.6%

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

7. Simplified 29.6%

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

   1. expm1-log1p-u 1.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-t\right)\right)} \]

   2. expm1-undefine 1.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]

9. Applied egg-rr 1.5%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]
10. Step-by-step derivation

    1. sub-neg 1.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} + \left(-1\right)} \]

    2. log1p-undefine 1.5%

       \[\leadsto e^{\color{blue}{\log \left(1 + \left(-t\right)\right)}} + \left(-1\right) \]

    3. rem-exp-log 29.7%

       \[\leadsto \color{blue}{\left(1 + \left(-t\right)\right)} + \left(-1\right) \]

    4. unsub-neg 29.7%

       \[\leadsto \color{blue}{\left(1 - t\right)} + \left(-1\right) \]

    5. metadata-eval 29.7%

       \[\leadsto \left(1 - t\right) + \color{blue}{-1} \]

11. Simplified 29.7%

    \[\leadsto \color{blue}{\left(1 - t\right) + -1} \]

12. Taylor expanded in t around 0 2.5%

    \[\leadsto \color{blue}{1} + -1 \]
13. Step-by-step derivation

    1. metadata-eval 2.5%

       \[\leadsto \color{blue}{0} \]

14. Applied egg-rr 2.5%

    \[\leadsto \color{blue}{0} \]
15. Add Preprocessing

Developer Target 1: 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 2024132 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (log (+ x y)) (+ (- (log z) t) (* (- a 1/2) (log t)))))

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