Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 23.1s

Alternatives: 17

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. associate--l+ 99.6%

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

   3. associate-+r+ 99.6%

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

   4. +-commutative 99.6%

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

   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. Final simplification 99.6%

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

Alternative 2: 59.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot \left(a - 0.5\right)\\ t_2 := \log z + \log \left(x + y\right)\\ t_3 := \log y + \left(\log z + t\_1\right)\\ \mathbf{if}\;t\_2 \leq -750:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 700:\\ \;\;\;\;\log \left(z \cdot y\right) + \left(t\_1 - t\right)\\ \mathbf{elif}\;t\_2 \leq 950 \lor \neg \left(t\_2 \leq 1050\right):\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) (- a 0.5)))
        (t_2 (+ (log z) (log (+ x y))))
        (t_3 (+ (log y) (+ (log z) t_1))))
   (if (<= t_2 -750.0)
     t_3
     (if (<= t_2 700.0)
       (+ (log (* z y)) (- t_1 t))
       (if (or (<= t_2 950.0) (not (<= t_2 1050.0)))
         t_3
         (- (+ (log z) (log 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 t_3 = log(y) + (log(z) + t_1);
	double tmp;
	if (t_2 <= -750.0) {
		tmp = t_3;
	} else if (t_2 <= 700.0) {
		tmp = log((z * y)) + (t_1 - t);
	} else if ((t_2 <= 950.0) || !(t_2 <= 1050.0)) {
		tmp = t_3;
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = log(t) * (a - 0.5d0)
    t_2 = log(z) + log((x + y))
    t_3 = log(y) + (log(z) + t_1)
    if (t_2 <= (-750.0d0)) then
        tmp = t_3
    else if (t_2 <= 700.0d0) then
        tmp = log((z * y)) + (t_1 - t)
    else if ((t_2 <= 950.0d0) .or. (.not. (t_2 <= 1050.0d0))) then
        tmp = t_3
    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 t_1 = Math.log(t) * (a - 0.5);
	double t_2 = Math.log(z) + Math.log((x + y));
	double t_3 = Math.log(y) + (Math.log(z) + t_1);
	double tmp;
	if (t_2 <= -750.0) {
		tmp = t_3;
	} else if (t_2 <= 700.0) {
		tmp = Math.log((z * y)) + (t_1 - t);
	} else if ((t_2 <= 950.0) || !(t_2 <= 1050.0)) {
		tmp = t_3;
	} else {
		tmp = (Math.log(z) + Math.log(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))
	t_3 = math.log(y) + (math.log(z) + t_1)
	tmp = 0
	if t_2 <= -750.0:
		tmp = t_3
	elif t_2 <= 700.0:
		tmp = math.log((z * y)) + (t_1 - t)
	elif (t_2 <= 950.0) or not (t_2 <= 1050.0):
		tmp = t_3
	else:
		tmp = (math.log(z) + math.log(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)))
	t_3 = Float64(log(y) + Float64(log(z) + t_1))
	tmp = 0.0
	if (t_2 <= -750.0)
		tmp = t_3;
	elseif (t_2 <= 700.0)
		tmp = Float64(log(Float64(z * y)) + Float64(t_1 - t));
	elseif ((t_2 <= 950.0) || !(t_2 <= 1050.0))
		tmp = t_3;
	else
		tmp = Float64(Float64(log(z) + log(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));
	t_3 = log(y) + (log(z) + t_1);
	tmp = 0.0;
	if (t_2 <= -750.0)
		tmp = t_3;
	elseif (t_2 <= 700.0)
		tmp = log((z * y)) + (t_1 - t);
	elseif ((t_2 <= 950.0) || ~((t_2 <= 1050.0)))
		tmp = t_3;
	else
		tmp = (log(z) + log(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]}, Block[{t$95$3 = N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -750.0], t$95$3, If[LessEqual[t$95$2, 700.0], N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] + N[(t$95$1 - t), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$2, 950.0], N[Not[LessEqual[t$95$2, 1050.0]], $MachinePrecision]], t$95$3, N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 700 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 950 or 1050 < (+.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%

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

      2. associate--l+ 99.7%

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

      3. sub-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-undefine 99.7%

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

      8. sub-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. distribute-neg-in 99.7%

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

      11. metadata-eval 99.7%

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

      12. metadata-eval 99.7%

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

      13. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 62.4%

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

   6. Taylor expanded in t around 0 43.8%

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

      1. associate--l+ 43.8%

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

   8. Simplified 43.8%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 700

   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 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 65.3%

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

      1. *-un-lft-identity 65.3%

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

      2. sum-log 62.1%

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

   7. Applied egg-rr 62.1%

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

      1. *-lft-identity 62.1%

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

      2. *-commutative 62.1%

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

   9. Simplified 62.1%

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

   if 950 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 1050

   1. Initial program 99.9%

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

      1. remove-double-neg 99.9%

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

      2. associate--l+ 99.7%

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

      3. remove-double-neg 99.7%

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

      4. sub-neg 99.7%

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

      5. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 41.7%

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

      1. associate--l+ 41.7%

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

      2. remove-double-neg 41.7%

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

      3. log-rec 41.7%

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

      4. mul-1-neg 41.7%

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

      5. mul-1-neg 41.7%

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

      6. log-rec 41.7%

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

      7. remove-double-neg 41.7%

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

      8. sub-neg 41.7%

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

      9. metadata-eval 41.7%

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

   7. Simplified 41.7%

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

   8. Taylor expanded in a around inf 41.7%

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

      1. *-commutative 41.7%

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

   10. Simplified 41.7%

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

   11. Taylor expanded in a around 0 41.7%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(x + y\right) \leq -750:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{elif}\;\log z + \log \left(x + y\right) \leq 700:\\ \;\;\;\;\log \left(z \cdot y\right) + \left(\log t \cdot \left(a - 0.5\right) - t\right)\\ \mathbf{elif}\;\log z + \log \left(x + y\right) \leq 950 \lor \neg \left(\log z + \log \left(x + y\right) \leq 1050\right):\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot \left(a - 0.5\right)\\ t_2 := \log z + \log \left(x + y\right)\\ t_3 := \log y + \left(\log z + t\_1\right)\\ \mathbf{if}\;t\_2 \leq -750:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 700:\\ \;\;\;\;\log \left(z \cdot y\right) + \left(t\_1 - t\right)\\ \mathbf{elif}\;t\_2 \leq 950 \lor \neg \left(t\_2 \leq 1050\right):\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\log y + \left(\left(\log z + -0.5 \cdot \log t\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) (- a 0.5)))
        (t_2 (+ (log z) (log (+ x y))))
        (t_3 (+ (log y) (+ (log z) t_1))))
   (if (<= t_2 -750.0)
     t_3
     (if (<= t_2 700.0)
       (+ (log (* z y)) (- t_1 t))
       (if (or (<= t_2 950.0) (not (<= t_2 1050.0)))
         t_3
         (+ (log y) (- (+ (log z) (* -0.5 (log t))) 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 t_3 = log(y) + (log(z) + t_1);
	double tmp;
	if (t_2 <= -750.0) {
		tmp = t_3;
	} else if (t_2 <= 700.0) {
		tmp = log((z * y)) + (t_1 - t);
	} else if ((t_2 <= 950.0) || !(t_2 <= 1050.0)) {
		tmp = t_3;
	} else {
		tmp = log(y) + ((log(z) + (-0.5 * 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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = log(t) * (a - 0.5d0)
    t_2 = log(z) + log((x + y))
    t_3 = log(y) + (log(z) + t_1)
    if (t_2 <= (-750.0d0)) then
        tmp = t_3
    else if (t_2 <= 700.0d0) then
        tmp = log((z * y)) + (t_1 - t)
    else if ((t_2 <= 950.0d0) .or. (.not. (t_2 <= 1050.0d0))) then
        tmp = t_3
    else
        tmp = log(y) + ((log(z) + ((-0.5d0) * 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 t_1 = Math.log(t) * (a - 0.5);
	double t_2 = Math.log(z) + Math.log((x + y));
	double t_3 = Math.log(y) + (Math.log(z) + t_1);
	double tmp;
	if (t_2 <= -750.0) {
		tmp = t_3;
	} else if (t_2 <= 700.0) {
		tmp = Math.log((z * y)) + (t_1 - t);
	} else if ((t_2 <= 950.0) || !(t_2 <= 1050.0)) {
		tmp = t_3;
	} else {
		tmp = Math.log(y) + ((Math.log(z) + (-0.5 * Math.log(t))) - 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))
	t_3 = math.log(y) + (math.log(z) + t_1)
	tmp = 0
	if t_2 <= -750.0:
		tmp = t_3
	elif t_2 <= 700.0:
		tmp = math.log((z * y)) + (t_1 - t)
	elif (t_2 <= 950.0) or not (t_2 <= 1050.0):
		tmp = t_3
	else:
		tmp = math.log(y) + ((math.log(z) + (-0.5 * math.log(t))) - 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)))
	t_3 = Float64(log(y) + Float64(log(z) + t_1))
	tmp = 0.0
	if (t_2 <= -750.0)
		tmp = t_3;
	elseif (t_2 <= 700.0)
		tmp = Float64(log(Float64(z * y)) + Float64(t_1 - t));
	elseif ((t_2 <= 950.0) || !(t_2 <= 1050.0))
		tmp = t_3;
	else
		tmp = Float64(log(y) + Float64(Float64(log(z) + Float64(-0.5 * log(t))) - 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));
	t_3 = log(y) + (log(z) + t_1);
	tmp = 0.0;
	if (t_2 <= -750.0)
		tmp = t_3;
	elseif (t_2 <= 700.0)
		tmp = log((z * y)) + (t_1 - t);
	elseif ((t_2 <= 950.0) || ~((t_2 <= 1050.0)))
		tmp = t_3;
	else
		tmp = log(y) + ((log(z) + (-0.5 * log(t))) - 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]}, Block[{t$95$3 = N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -750.0], t$95$3, If[LessEqual[t$95$2, 700.0], N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] + N[(t$95$1 - t), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$2, 950.0], N[Not[LessEqual[t$95$2, 1050.0]], $MachinePrecision]], t$95$3, N[(N[Log[y], $MachinePrecision] + N[(N[(N[Log[z], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 700 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 950 or 1050 < (+.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%

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

      2. associate--l+ 99.7%

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

      3. sub-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-undefine 99.7%

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

      8. sub-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. distribute-neg-in 99.7%

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

      11. metadata-eval 99.7%

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

      12. metadata-eval 99.7%

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

      13. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 62.4%

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

   6. Taylor expanded in t around 0 43.8%

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

      1. associate--l+ 43.8%

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

   8. Simplified 43.8%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 700

   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 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 65.3%

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

      1. *-un-lft-identity 65.3%

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

      2. sum-log 62.1%

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

   7. Applied egg-rr 62.1%

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

      1. *-lft-identity 62.1%

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

      2. *-commutative 62.1%

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

   9. Simplified 62.1%

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

   if 950 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 1050

   1. Initial program 99.9%

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

      1. remove-double-neg 99.9%

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

      2. associate--l+ 99.7%

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

      3. remove-double-neg 99.7%

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

      4. sub-neg 99.7%

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

      5. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 41.7%

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

      1. associate--l+ 41.7%

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

      2. remove-double-neg 41.7%

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

      3. log-rec 41.7%

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

      4. mul-1-neg 41.7%

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

      5. mul-1-neg 41.7%

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

      6. log-rec 41.7%

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

      7. remove-double-neg 41.7%

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

      8. sub-neg 41.7%

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

      9. metadata-eval 41.7%

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

   7. Simplified 41.7%

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

   8. Taylor expanded in a around 0 41.7%

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

      1. *-commutative 41.7%

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

   10. Simplified 41.7%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(x + y\right) \leq -750:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{elif}\;\log z + \log \left(x + y\right) \leq 700:\\ \;\;\;\;\log \left(z \cdot y\right) + \left(\log t \cdot \left(a - 0.5\right) - t\right)\\ \mathbf{elif}\;\log z + \log \left(x + y\right) \leq 950 \lor \neg \left(\log z + \log \left(x + y\right) \leq 1050\right):\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log y + \left(\left(\log z + -0.5 \cdot \log t\right) - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log z + \log \left(x + y\right)\\ t_2 := \log y + \left(\log z + a \cdot \log t\right)\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 700:\\ \;\;\;\;\log \left(z \cdot y\right) + \left(\log t \cdot \left(a - 0.5\right) - t\right)\\ \mathbf{elif}\;t\_1 \leq 950 \lor \neg \left(t\_1 \leq 1050\right):\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log z) (log (+ x y))))
        (t_2 (+ (log y) (+ (log z) (* a (log t))))))
   (if (<= t_1 -750.0)
     t_2
     (if (<= t_1 700.0)
       (+ (log (* z y)) (- (* (log t) (- a 0.5)) t))
       (if (or (<= t_1 950.0) (not (<= t_1 1050.0)))
         t_2
         (- (+ (log z) (log y)) 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) + (a * log(t)));
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 700.0) {
		tmp = log((z * y)) + ((log(t) * (a - 0.5)) - t);
	} else if ((t_1 <= 950.0) || !(t_1 <= 1050.0)) {
		tmp = t_2;
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log(z) + log((x + y))
    t_2 = log(y) + (log(z) + (a * log(t)))
    if (t_1 <= (-750.0d0)) then
        tmp = t_2
    else if (t_1 <= 700.0d0) then
        tmp = log((z * y)) + ((log(t) * (a - 0.5d0)) - t)
    else if ((t_1 <= 950.0d0) .or. (.not. (t_1 <= 1050.0d0))) then
        tmp = t_2
    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 t_1 = Math.log(z) + Math.log((x + y));
	double t_2 = Math.log(y) + (Math.log(z) + (a * Math.log(t)));
	double tmp;
	if (t_1 <= -750.0) {
		tmp = t_2;
	} else if (t_1 <= 700.0) {
		tmp = Math.log((z * y)) + ((Math.log(t) * (a - 0.5)) - t);
	} else if ((t_1 <= 950.0) || !(t_1 <= 1050.0)) {
		tmp = t_2;
	} else {
		tmp = (Math.log(z) + Math.log(y)) - 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) + (a * math.log(t)))
	tmp = 0
	if t_1 <= -750.0:
		tmp = t_2
	elif t_1 <= 700.0:
		tmp = math.log((z * y)) + ((math.log(t) * (a - 0.5)) - t)
	elif (t_1 <= 950.0) or not (t_1 <= 1050.0):
		tmp = t_2
	else:
		tmp = (math.log(z) + math.log(y)) - 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(z) + Float64(a * log(t))))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 700.0)
		tmp = Float64(log(Float64(z * y)) + Float64(Float64(log(t) * Float64(a - 0.5)) - t));
	elseif ((t_1 <= 950.0) || !(t_1 <= 1050.0))
		tmp = t_2;
	else
		tmp = Float64(Float64(log(z) + log(y)) - 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) + (a * log(t)));
	tmp = 0.0;
	if (t_1 <= -750.0)
		tmp = t_2;
	elseif (t_1 <= 700.0)
		tmp = log((z * y)) + ((log(t) * (a - 0.5)) - t);
	elseif ((t_1 <= 950.0) || ~((t_1 <= 1050.0)))
		tmp = t_2;
	else
		tmp = (log(z) + log(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]}, Block[{t$95$2 = N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], t$95$2, If[LessEqual[t$95$1, 700.0], N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$1, 950.0], N[Not[LessEqual[t$95$1, 1050.0]], $MachinePrecision]], t$95$2, N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 700 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 950 or 1050 < (+.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. remove-double-neg 99.7%

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

      2. associate--l+ 99.7%

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

      3. remove-double-neg 99.7%

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

      4. sub-neg 99.7%

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

      5. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 62.4%

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

      1. associate--l+ 62.4%

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

      2. remove-double-neg 62.4%

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

      3. log-rec 62.4%

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

      4. mul-1-neg 62.4%

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

      5. mul-1-neg 62.4%

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

      6. log-rec 62.4%

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

      7. remove-double-neg 62.4%

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

      8. sub-neg 62.4%

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

      9. metadata-eval 62.4%

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

   7. Simplified 62.4%

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

   8. Taylor expanded in a around inf 52.2%

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

      1. *-commutative 52.2%

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

   10. Simplified 52.2%

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

   11. Taylor expanded in t around 0 33.5%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 700

   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 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 65.3%

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

      1. *-un-lft-identity 65.3%

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

      2. sum-log 62.1%

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

   7. Applied egg-rr 62.1%

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

      1. *-lft-identity 62.1%

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

      2. *-commutative 62.1%

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

   9. Simplified 62.1%

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

   if 950 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 1050

   1. Initial program 99.9%

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

      1. remove-double-neg 99.9%

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

      2. associate--l+ 99.7%

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

      3. remove-double-neg 99.7%

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

      4. sub-neg 99.7%

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

      5. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 41.7%

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

      1. associate--l+ 41.7%

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

      2. remove-double-neg 41.7%

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

      3. log-rec 41.7%

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

      4. mul-1-neg 41.7%

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

      5. mul-1-neg 41.7%

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

      6. log-rec 41.7%

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

      7. remove-double-neg 41.7%

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

      8. sub-neg 41.7%

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

      9. metadata-eval 41.7%

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

   7. Simplified 41.7%

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

   8. Taylor expanded in a around inf 41.7%

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

      1. *-commutative 41.7%

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

   10. Simplified 41.7%

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

   11. Taylor expanded in a around 0 41.7%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(x + y\right) \leq -750:\\ \;\;\;\;\log y + \left(\log z + a \cdot \log t\right)\\ \mathbf{elif}\;\log z + \log \left(x + y\right) \leq 700:\\ \;\;\;\;\log \left(z \cdot y\right) + \left(\log t \cdot \left(a - 0.5\right) - t\right)\\ \mathbf{elif}\;\log z + \log \left(x + y\right) \leq 950 \lor \neg \left(\log z + \log \left(x + y\right) \leq 1050\right):\\ \;\;\;\;\log y + \left(\log z + a \cdot \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right)\\ t_2 := \log z + t\_1\\ t_3 := \log y + a \cdot \log t\\ \mathbf{if}\;t\_2 \leq -750:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 700:\\ \;\;\;\;\log \left(z \cdot y\right) + \left(\log t \cdot \left(a - 0.5\right) - t\right)\\ \mathbf{elif}\;t\_2 \leq 950:\\ \;\;\;\;a \cdot \left(\log t + \frac{t\_1}{a}\right)\\ \mathbf{elif}\;t\_2 \leq 1050:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ x y)))
        (t_2 (+ (log z) t_1))
        (t_3 (+ (log y) (* a (log t)))))
   (if (<= t_2 -750.0)
     t_3
     (if (<= t_2 700.0)
       (+ (log (* z y)) (- (* (log t) (- a 0.5)) t))
       (if (<= t_2 950.0)
         (* a (+ (log t) (/ t_1 a)))
         (if (<= t_2 1050.0) (- (+ (log z) (log y)) t) t_3))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y));
	double t_2 = log(z) + t_1;
	double t_3 = log(y) + (a * log(t));
	double tmp;
	if (t_2 <= -750.0) {
		tmp = t_3;
	} else if (t_2 <= 700.0) {
		tmp = log((z * y)) + ((log(t) * (a - 0.5)) - t);
	} else if (t_2 <= 950.0) {
		tmp = a * (log(t) + (t_1 / a));
	} else if (t_2 <= 1050.0) {
		tmp = (log(z) + log(y)) - t;
	} else {
		tmp = t_3;
	}
	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) :: t_3
    real(8) :: tmp
    t_1 = log((x + y))
    t_2 = log(z) + t_1
    t_3 = log(y) + (a * log(t))
    if (t_2 <= (-750.0d0)) then
        tmp = t_3
    else if (t_2 <= 700.0d0) then
        tmp = log((z * y)) + ((log(t) * (a - 0.5d0)) - t)
    else if (t_2 <= 950.0d0) then
        tmp = a * (log(t) + (t_1 / a))
    else if (t_2 <= 1050.0d0) then
        tmp = (log(z) + log(y)) - t
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log((x + y));
	double t_2 = Math.log(z) + t_1;
	double t_3 = Math.log(y) + (a * Math.log(t));
	double tmp;
	if (t_2 <= -750.0) {
		tmp = t_3;
	} else if (t_2 <= 700.0) {
		tmp = Math.log((z * y)) + ((Math.log(t) * (a - 0.5)) - t);
	} else if (t_2 <= 950.0) {
		tmp = a * (Math.log(t) + (t_1 / a));
	} else if (t_2 <= 1050.0) {
		tmp = (Math.log(z) + Math.log(y)) - t;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log((x + y))
	t_2 = math.log(z) + t_1
	t_3 = math.log(y) + (a * math.log(t))
	tmp = 0
	if t_2 <= -750.0:
		tmp = t_3
	elif t_2 <= 700.0:
		tmp = math.log((z * y)) + ((math.log(t) * (a - 0.5)) - t)
	elif t_2 <= 950.0:
		tmp = a * (math.log(t) + (t_1 / a))
	elif t_2 <= 1050.0:
		tmp = (math.log(z) + math.log(y)) - t
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	t_2 = Float64(log(z) + t_1)
	t_3 = Float64(log(y) + Float64(a * log(t)))
	tmp = 0.0
	if (t_2 <= -750.0)
		tmp = t_3;
	elseif (t_2 <= 700.0)
		tmp = Float64(log(Float64(z * y)) + Float64(Float64(log(t) * Float64(a - 0.5)) - t));
	elseif (t_2 <= 950.0)
		tmp = Float64(a * Float64(log(t) + Float64(t_1 / a)));
	elseif (t_2 <= 1050.0)
		tmp = Float64(Float64(log(z) + log(y)) - t);
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log((x + y));
	t_2 = log(z) + t_1;
	t_3 = log(y) + (a * log(t));
	tmp = 0.0;
	if (t_2 <= -750.0)
		tmp = t_3;
	elseif (t_2 <= 700.0)
		tmp = log((z * y)) + ((log(t) * (a - 0.5)) - t);
	elseif (t_2 <= 950.0)
		tmp = a * (log(t) + (t_1 / a));
	elseif (t_2 <= 1050.0)
		tmp = (log(z) + log(y)) - t;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[z], $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Log[y], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -750.0], t$95$3, If[LessEqual[t$95$2, 700.0], N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 950.0], N[(a * N[(N[Log[t], $MachinePrecision] + N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1050.0], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750 or 1050 < (+.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%

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

      2. associate--l+ 99.7%

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

      3. sub-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-undefine 99.7%

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

      8. sub-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. distribute-neg-in 99.7%

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

      11. metadata-eval 99.7%

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

      12. metadata-eval 99.7%

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

      13. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around inf 49.1%

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

      1. *-commutative 49.1%

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

   7. Simplified 49.1%

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

   8. Taylor expanded in x around 0 31.5%

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

      1. +-commutative 31.5%

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

   10. Simplified 31.5%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 700

   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 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 65.3%

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

      1. *-un-lft-identity 65.3%

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

      2. sum-log 62.1%

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

   7. Applied egg-rr 62.1%

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

      1. *-lft-identity 62.1%

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

      2. *-commutative 62.1%

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

   9. Simplified 62.1%

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

   if 700 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 950

   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%

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

      2. associate--l+ 99.7%

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

      3. sub-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-undefine 99.7%

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

      8. sub-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. distribute-neg-in 99.7%

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

      11. metadata-eval 99.7%

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

      12. metadata-eval 99.7%

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

      13. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around inf 56.5%

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

      1. *-commutative 56.5%

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

   7. Simplified 56.5%

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

   8. Taylor expanded in a around inf 56.5%

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

      1. +-commutative 56.5%

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

   10. Simplified 56.5%

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

   if 950 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 1050

   1. Initial program 99.9%

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

      1. remove-double-neg 99.9%

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

      2. associate--l+ 99.7%

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

      3. remove-double-neg 99.7%

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

      4. sub-neg 99.7%

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

      5. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 41.7%

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

      1. associate--l+ 41.7%

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

      2. remove-double-neg 41.7%

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

      3. log-rec 41.7%

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

      4. mul-1-neg 41.7%

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

      5. mul-1-neg 41.7%

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

      6. log-rec 41.7%

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

      7. remove-double-neg 41.7%

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

      8. sub-neg 41.7%

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

      9. metadata-eval 41.7%

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

   7. Simplified 41.7%

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

   8. Taylor expanded in a around inf 41.7%

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

      1. *-commutative 41.7%

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

   10. Simplified 41.7%

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

   11. Taylor expanded in a around 0 41.7%

       \[\leadsto \color{blue}{\left(\log y + \log z\right) - t} \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z + \log \left(x + y\right) \leq -750:\\ \;\;\;\;\log y + a \cdot \log t\\ \mathbf{elif}\;\log z + \log \left(x + y\right) \leq 700:\\ \;\;\;\;\log \left(z \cdot y\right) + \left(\log t \cdot \left(a - 0.5\right) - t\right)\\ \mathbf{elif}\;\log z + \log \left(x + y\right) \leq 950:\\ \;\;\;\;a \cdot \left(\log t + \frac{\log \left(x + y\right)}{a}\right)\\ \mathbf{elif}\;\log z + \log \left(x + y\right) \leq 1050:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y + a \cdot \log t\\ \end{array} \]
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 99.6%

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

   1. associate-+l- 99.6%

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

   2. associate--l+ 99.6%

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

   3. sub-neg 99.6%

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

   4. +-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. distribute-rgt-neg-in 99.6%

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

   7. fma-undefine 99.6%

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

   8. sub-neg 99.6%

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

   9. +-commutative 99.6%

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

   10. distribute-neg-in 99.6%

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

   11. metadata-eval 99.6%

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

   12. metadata-eval 99.6%

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

   13. unsub-neg 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 7: 68.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.43:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log y + \left(\left(\log z + a \cdot \log t\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 0.43)
   (+ (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 <= 0.43) {
		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 <= 0.43d0) 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 <= 0.43) {
		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 <= 0.43:
		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 <= 0.43)
		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 <= 0.43)
		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, 0.43], 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 < 0.429999999999999993

   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%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. fma-undefine 99.3%

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

      8. sub-neg 99.3%

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

      9. +-commutative 99.3%

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

      10. distribute-neg-in 99.3%

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

      11. metadata-eval 99.3%

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

      12. metadata-eval 99.3%

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

      13. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 62.7%

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

   6. Taylor expanded in t around 0 62.2%

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

      1. associate--l+ 62.2%

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

   8. Simplified 62.2%

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

   if 0.429999999999999993 < t

   1. Initial program 99.9%

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

      1. remove-double-neg 99.9%

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

      2. associate--l+ 99.9%

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

      3. remove-double-neg 99.9%

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

      4. sub-neg 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 64.4%

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

      1. associate--l+ 64.4%

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

      2. remove-double-neg 64.4%

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

      3. log-rec 64.4%

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

      4. mul-1-neg 64.4%

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

      5. mul-1-neg 64.4%

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

      6. log-rec 64.4%

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

      7. remove-double-neg 64.4%

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

      8. sub-neg 64.4%

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

      9. metadata-eval 64.4%

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

   7. Simplified 64.4%

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

   8. Taylor expanded in a around inf 64.0%

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

      1. *-commutative 64.0%

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

   10. Simplified 64.0%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.43:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log y + \left(\left(\log z + a \cdot \log t\right) - t\right)\\ \end{array} \]
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 99.6%

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

   1. remove-double-neg 99.6%

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

   2. associate--l+ 99.6%

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

   3. remove-double-neg 99.6%

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

   4. sub-neg 99.6%

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

   5. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 9: 69.2% 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 99.6%

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

   1. remove-double-neg 99.6%

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

   2. associate--l+ 99.6%

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

   3. remove-double-neg 99.6%

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

   4. sub-neg 99.6%

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

   5. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 63.6%

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

   1. associate--l+ 63.6%

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

   2. remove-double-neg 63.6%

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

   3. log-rec 63.6%

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

   4. mul-1-neg 63.6%

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

   5. mul-1-neg 63.6%

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

   6. log-rec 63.6%

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

   7. remove-double-neg 63.6%

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

   8. sub-neg 63.6%

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

   9. metadata-eval 63.6%

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

7. Simplified 63.6%

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

8. Final simplification 63.6%

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

Alternative 10: 52.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{-294}:\\ \;\;\;\;\log y + a \cdot \log t\\ \mathbf{elif}\;t \leq 17500000000000:\\ \;\;\;\;\log \left(z \cdot y\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 3.4e-294)
   (+ (log y) (* a (log t)))
   (if (<= t 17500000000000.0)
     (+ (log (* z y)) (* (log t) (- a 0.5)))
     (- (+ (log z) (log y)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 3.4e-294) {
		tmp = log(y) + (a * log(t));
	} else if (t <= 17500000000000.0) {
		tmp = log((z * y)) + (log(t) * (a - 0.5));
	} 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 (t <= 3.4d-294) then
        tmp = log(y) + (a * log(t))
    else if (t <= 17500000000000.0d0) then
        tmp = log((z * y)) + (log(t) * (a - 0.5d0))
    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 (t <= 3.4e-294) {
		tmp = Math.log(y) + (a * Math.log(t));
	} else if (t <= 17500000000000.0) {
		tmp = Math.log((z * y)) + (Math.log(t) * (a - 0.5));
	} else {
		tmp = (Math.log(z) + Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 3.4e-294:
		tmp = math.log(y) + (a * math.log(t))
	elif t <= 17500000000000.0:
		tmp = math.log((z * y)) + (math.log(t) * (a - 0.5))
	else:
		tmp = (math.log(z) + math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 3.4e-294)
		tmp = Float64(log(y) + Float64(a * log(t)));
	elseif (t <= 17500000000000.0)
		tmp = Float64(log(Float64(z * y)) + Float64(log(t) * Float64(a - 0.5)));
	else
		tmp = Float64(Float64(log(z) + log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 3.4e-294)
		tmp = log(y) + (a * log(t));
	elseif (t <= 17500000000000.0)
		tmp = log((z * y)) + (log(t) * (a - 0.5));
	else
		tmp = (log(z) + log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 3.4e-294], N[(N[Log[y], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 17500000000000.0], N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.39999999999999981e-294

   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+ 99.3%

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

      3. sub-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. fma-undefine 99.3%

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

      8. sub-neg 99.3%

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

      9. +-commutative 99.3%

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

      10. distribute-neg-in 99.3%

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

      11. metadata-eval 99.3%

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

      12. metadata-eval 99.3%

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

      13. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in a around inf 68.0%

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

      1. *-commutative 68.0%

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

   7. Simplified 68.0%

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

   8. Taylor expanded in x around 0 48.4%

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

      1. +-commutative 48.4%

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

   10. Simplified 48.4%

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

   if 3.39999999999999981e-294 < t < 1.75e13

   1. Initial program 99.2%

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

      1. associate-+l- 99.3%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. fma-undefine 99.3%

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

      8. sub-neg 99.3%

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

      9. +-commutative 99.3%

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

      10. distribute-neg-in 99.3%

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

      11. metadata-eval 99.3%

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

      12. metadata-eval 99.3%

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

      13. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 62.3%

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

      1. sub-neg 62.3%

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

      2. sum-log 46.9%

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

      3. +-commutative 46.9%

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

      4. fma-define 46.9%

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

   7. Applied egg-rr 46.9%

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

      1. sub-neg 46.9%

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

      2. *-commutative 46.9%

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

   9. Simplified 46.9%

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

   10. Taylor expanded in t around 0 45.3%

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

   if 1.75e13 < t

   1. Initial program 99.9%

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

      1. remove-double-neg 99.9%

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

      2. associate--l+ 99.9%

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

      3. remove-double-neg 99.9%

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

      4. sub-neg 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 65.0%

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

      1. associate--l+ 65.0%

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

      2. remove-double-neg 65.0%

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

      3. log-rec 65.0%

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

      4. mul-1-neg 65.0%

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

      5. mul-1-neg 65.0%

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

      6. log-rec 65.0%

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

      7. remove-double-neg 65.0%

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

      8. sub-neg 65.0%

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

      9. metadata-eval 65.0%

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

   7. Simplified 65.0%

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

   8. Taylor expanded in a around inf 65.0%

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

      1. *-commutative 65.0%

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

   10. Simplified 65.0%

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

   11. Taylor expanded in a around 0 53.7%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{-294}:\\ \;\;\;\;\log y + a \cdot \log t\\ \mathbf{elif}\;t \leq 17500000000000:\\ \;\;\;\;\log \left(z \cdot y\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 14000000000000:\\ \;\;\;\;a \cdot \left(\log t + \frac{\log \left(x + y\right)}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 14000000000000.0)
   (* a (+ (log t) (/ (log (+ x y)) a)))
   (- (+ (log z) (log y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 14000000000000.0) {
		tmp = a * (log(t) + (log((x + y)) / a));
	} else {
		tmp = (log(z) + log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 14000000000000.0d0) then
        tmp = a * (log(t) + (log((x + y)) / a))
    else
        tmp = (log(z) + log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 14000000000000.0) {
		tmp = a * (Math.log(t) + (Math.log((x + y)) / a));
	} else {
		tmp = (Math.log(z) + Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 14000000000000.0:
		tmp = a * (math.log(t) + (math.log((x + y)) / a))
	else:
		tmp = (math.log(z) + math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 14000000000000.0)
		tmp = Float64(a * Float64(log(t) + Float64(log(Float64(x + y)) / a)));
	else
		tmp = Float64(Float64(log(z) + log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 14000000000000.0)
		tmp = a * (log(t) + (log((x + y)) / a));
	else
		tmp = (log(z) + log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 14000000000000.0], N[(a * N[(N[Log[t], $MachinePrecision] + N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.4e13

   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%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. fma-undefine 99.3%

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

      8. sub-neg 99.3%

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

      9. +-commutative 99.3%

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

      10. distribute-neg-in 99.3%

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

      11. metadata-eval 99.3%

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

      12. metadata-eval 99.3%

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

      13. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in a around inf 57.0%

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

      1. *-commutative 57.0%

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

   7. Simplified 57.0%

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

   8. Taylor expanded in a around inf 57.0%

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

      1. +-commutative 57.0%

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

   10. Simplified 57.0%

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

   if 1.4e13 < t

   1. Initial program 99.9%

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

      1. remove-double-neg 99.9%

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

      2. associate--l+ 99.9%

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

      3. remove-double-neg 99.9%

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

      4. sub-neg 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 65.0%

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

      1. associate--l+ 65.0%

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

      2. remove-double-neg 65.0%

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

      3. log-rec 65.0%

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

      4. mul-1-neg 65.0%

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

      5. mul-1-neg 65.0%

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

      6. log-rec 65.0%

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

      7. remove-double-neg 65.0%

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

      8. sub-neg 65.0%

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

      9. metadata-eval 65.0%

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

   7. Simplified 65.0%

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

   8. Taylor expanded in a around inf 65.0%

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

      1. *-commutative 65.0%

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

   10. Simplified 65.0%

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

   11. Taylor expanded in a around 0 53.7%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 14000000000000:\\ \;\;\;\;a \cdot \left(\log t + \frac{\log \left(x + y\right)}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.85 \cdot 10^{+14}:\\ \;\;\;\;\log y + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.85e+14) (+ (log y) (* a (log t))) (- (log (+ x y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.85e+14) {
		tmp = log(y) + (a * log(t));
	} else {
		tmp = log((x + y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 1.85d+14) then
        tmp = log(y) + (a * log(t))
    else
        tmp = log((x + y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.85e+14) {
		tmp = Math.log(y) + (a * Math.log(t));
	} else {
		tmp = Math.log((x + y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.85e+14:
		tmp = math.log(y) + (a * math.log(t))
	else:
		tmp = math.log((x + y)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.85e+14)
		tmp = Float64(log(y) + Float64(a * log(t)));
	else
		tmp = Float64(log(Float64(x + y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 1.85e+14)
		tmp = log(y) + (a * log(t));
	else
		tmp = log((x + y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.85e+14], N[(N[Log[y], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.85e14

   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%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. fma-undefine 99.3%

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

      8. sub-neg 99.3%

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

      9. +-commutative 99.3%

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

      10. distribute-neg-in 99.3%

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

      11. metadata-eval 99.3%

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

      12. metadata-eval 99.3%

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

      13. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in a around inf 57.0%

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

      1. *-commutative 57.0%

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

   7. Simplified 57.0%

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

   8. Taylor expanded in x around 0 42.0%

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

      1. +-commutative 42.0%

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

   10. Simplified 42.0%

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

   if 1.85e14 < t

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. associate--l+ 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-undefine 99.9%

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

      8. sub-neg 99.9%

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

      9. +-commutative 99.9%

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

      10. distribute-neg-in 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 76.7%

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

      1. neg-mul-1 76.7%

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

   7. Simplified 76.7%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.85 \cdot 10^{+14}:\\ \;\;\;\;\log y + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 48.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 33000000000:\\ \;\;\;\;\log y + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 33000000000.0) (+ (log y) (* a (log t))) (- (+ (log z) (log y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 33000000000.0) {
		tmp = log(y) + (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 (t <= 33000000000.0d0) then
        tmp = log(y) + (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 (t <= 33000000000.0) {
		tmp = Math.log(y) + (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 t <= 33000000000.0:
		tmp = math.log(y) + (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 (t <= 33000000000.0)
		tmp = Float64(log(y) + Float64(a * log(t)));
	else
		tmp = Float64(Float64(log(z) + log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 33000000000.0)
		tmp = log(y) + (a * log(t));
	else
		tmp = (log(z) + log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 33000000000.0], N[(N[Log[y], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.3e10

   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%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. fma-undefine 99.3%

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

      8. sub-neg 99.3%

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

      9. +-commutative 99.3%

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

      10. distribute-neg-in 99.3%

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

      11. metadata-eval 99.3%

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

      12. metadata-eval 99.3%

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

      13. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in a around inf 57.0%

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

      1. *-commutative 57.0%

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

   7. Simplified 57.0%

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

   8. Taylor expanded in x around 0 42.0%

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

      1. +-commutative 42.0%

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

   10. Simplified 42.0%

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

   if 3.3e10 < t

   1. Initial program 99.9%

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

      1. remove-double-neg 99.9%

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

      2. associate--l+ 99.9%

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

      3. remove-double-neg 99.9%

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

      4. sub-neg 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 65.0%

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

      1. associate--l+ 65.0%

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

      2. remove-double-neg 65.0%

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

      3. log-rec 65.0%

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

      4. mul-1-neg 65.0%

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

      5. mul-1-neg 65.0%

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

      6. log-rec 65.0%

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

      7. remove-double-neg 65.0%

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

      8. sub-neg 65.0%

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

      9. metadata-eval 65.0%

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

   7. Simplified 65.0%

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

   8. Taylor expanded in a around inf 65.0%

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

      1. *-commutative 65.0%

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

   10. Simplified 65.0%

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

   11. Taylor expanded in a around 0 53.7%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 33000000000:\\ \;\;\;\;\log y + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.6 \cdot 10^{+101} \lor \neg \left(a \leq 1.7 \cdot 10^{+17}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -5.6e+101) (not (<= a 1.7e+17)))
   (* a (log t))
   (- (log (+ x y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -5.6e+101) || !(a <= 1.7e+17)) {
		tmp = a * log(t);
	} else {
		tmp = log((x + y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-5.6d+101)) .or. (.not. (a <= 1.7d+17))) then
        tmp = a * log(t)
    else
        tmp = log((x + y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -5.6e+101) || !(a <= 1.7e+17)) {
		tmp = a * Math.log(t);
	} else {
		tmp = Math.log((x + y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -5.6e+101) or not (a <= 1.7e+17):
		tmp = a * math.log(t)
	else:
		tmp = math.log((x + y)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -5.6e+101) || !(a <= 1.7e+17))
		tmp = Float64(a * log(t));
	else
		tmp = Float64(log(Float64(x + y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -5.6e+101) || ~((a <= 1.7e+17)))
		tmp = a * log(t);
	else
		tmp = log((x + y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -5.6e+101], N[Not[LessEqual[a, 1.7e+17]], $MachinePrecision]], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.59999999999999962e101 or 1.7e17 < 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%

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

      2. associate--l+ 99.7%

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

      3. sub-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-undefine 99.7%

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

      8. sub-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. distribute-neg-in 99.7%

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

      11. metadata-eval 99.7%

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

      12. metadata-eval 99.7%

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

      13. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 67.8%

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

   6. Taylor expanded in a around inf 81.7%

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

      1. *-commutative 81.7%

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

   8. Simplified 81.7%

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

   if -5.59999999999999962e101 < a < 1.7e17

   1. Initial program 99.5%

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

      1. associate-+l- 99.5%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-undefine 99.5%

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

      8. sub-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. distribute-neg-in 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. unsub-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in t around inf 57.4%

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

      1. neg-mul-1 57.4%

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

   7. Simplified 57.4%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.6 \cdot 10^{+101} \lor \neg \left(a \leq 1.7 \cdot 10^{+17}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 40.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{-7}:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (if (<= t 2.1e-7) (log (+ x y)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.1e-7) {
		tmp = log((x + y));
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 2.1d-7) then
        tmp = log((x + y))
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.1e-7) {
		tmp = Math.log((x + y));
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 2.1e-7:
		tmp = math.log((x + y))
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.1e-7)
		tmp = log(Float64(x + y));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 2.1e-7)
		tmp = log((x + y));
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.1e-7], N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 2.1e-7

   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%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. fma-undefine 99.3%

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

      8. sub-neg 99.3%

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

      9. +-commutative 99.3%

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

      10. distribute-neg-in 99.3%

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

      11. metadata-eval 99.3%

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

      12. metadata-eval 99.3%

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

      13. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in a around inf 56.6%

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

      1. *-commutative 56.6%

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

   7. Simplified 56.6%

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

   8. Taylor expanded in a around 0 9.9%

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

      1. +-commutative 9.9%

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

   10. Simplified 9.9%

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

   if 2.1e-7 < t

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. associate--l+ 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-undefine 99.9%

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

      8. sub-neg 99.9%

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

      9. +-commutative 99.9%

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

      10. distribute-neg-in 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 64.2%

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

   6. Taylor expanded in t around inf 72.1%

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

      1. neg-mul-1 72.1%

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

   8. Simplified 72.1%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.1 \cdot 10^{-7}:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 62.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 15000000000:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 15000000000.0) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 15000000000.0) {
		tmp = a * log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 15000000000.0d0) then
        tmp = a * log(t)
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 15000000000.0) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 15000000000.0:
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 15000000000.0)
		tmp = Float64(a * log(t));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 15000000000.0)
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 15000000000.0], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 1.5e10

   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%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. fma-undefine 99.3%

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

      8. sub-neg 99.3%

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

      9. +-commutative 99.3%

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

      10. distribute-neg-in 99.3%

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

      11. metadata-eval 99.3%

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

      12. metadata-eval 99.3%

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

      13. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 62.2%

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

   6. Taylor expanded in a around inf 51.2%

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

      1. *-commutative 51.2%

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

   8. Simplified 51.2%

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

   if 1.5e10 < t

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. associate--l+ 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-undefine 99.9%

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

      8. sub-neg 99.9%

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

      9. +-commutative 99.9%

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

      10. distribute-neg-in 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 65.0%

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

   6. Taylor expanded in t around inf 76.7%

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

      1. neg-mul-1 76.7%

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

   8. Simplified 76.7%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 15000000000:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 37.6% accurate, 156.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-+l- 99.6%

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

   2. associate--l+ 99.6%

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

   3. sub-neg 99.6%

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

   4. +-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. distribute-rgt-neg-in 99.6%

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

   7. fma-undefine 99.6%

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

   8. sub-neg 99.6%

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

   9. +-commutative 99.6%

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

   10. distribute-neg-in 99.6%

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

   11. metadata-eval 99.6%

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

   12. metadata-eval 99.6%

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

   13. unsub-neg 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 63.5%

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

6. Taylor expanded in t around inf 38.6%

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

   1. neg-mul-1 38.6%

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

8. Simplified 38.6%

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

9. Final simplification 38.6%

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

Developer target: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024077 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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