Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 18.6s

Alternatives: 13

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = -3.2776348051353367e+127
  2. y = 8.547520619709137e+299
  3. z = 1.1310446734097206e+172
  4. t = 8.081398881647772e+50
  5. a = -1.2279689211646693e-284

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-+l- 99.5%

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

   2. associate--l+ 99.5%

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

   3. sub-neg 99.5%

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

   4. +-commutative 99.5%

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

   5. *-commutative 99.5%

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

   6. distribute-rgt-neg-in 99.5%

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

   7. fma-undefine 99.5%

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

   8. sub-neg 99.5%

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

   9. +-commutative 99.5%

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

   10. distribute-neg-in 99.5%

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

   11. metadata-eval 99.5%

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

   12. metadata-eval 99.5%

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

   13. unsub-neg 99.5%

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

3. Simplified 99.5%

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

Alternative 2: 91.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq -900 \lor \neg \left(t\_1 \leq 611\right):\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(\log t \cdot \left(a + -0.5\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (or (<= t_1 -900.0) (not (<= t_1 611.0)))
     (- (* (log t) (- a 0.5)) t)
     (+ (log (* (+ x y) z)) (- (* (log t) (+ a -0.5)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double tmp;
	if ((t_1 <= -900.0) || !(t_1 <= 611.0)) {
		tmp = (log(t) * (a - 0.5)) - t;
	} else {
		tmp = log(((x + y) * z)) + ((log(t) * (a + -0.5)) - t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log((x + y)) + log(z)
    if ((t_1 <= (-900.0d0)) .or. (.not. (t_1 <= 611.0d0))) then
        tmp = (log(t) * (a - 0.5d0)) - t
    else
        tmp = log(((x + y) * z)) + ((log(t) * (a + (-0.5d0))) - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log((x + y)) + Math.log(z);
	double tmp;
	if ((t_1 <= -900.0) || !(t_1 <= 611.0)) {
		tmp = (Math.log(t) * (a - 0.5)) - t;
	} else {
		tmp = Math.log(((x + y) * z)) + ((Math.log(t) * (a + -0.5)) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log((x + y)) + math.log(z)
	tmp = 0
	if (t_1 <= -900.0) or not (t_1 <= 611.0):
		tmp = (math.log(t) * (a - 0.5)) - t
	else:
		tmp = math.log(((x + y) * z)) + ((math.log(t) * (a + -0.5)) - t)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	tmp = 0.0
	if ((t_1 <= -900.0) || !(t_1 <= 611.0))
		tmp = Float64(Float64(log(t) * Float64(a - 0.5)) - t);
	else
		tmp = Float64(log(Float64(Float64(x + y) * z)) + Float64(Float64(log(t) * Float64(a + -0.5)) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log((x + y)) + log(z);
	tmp = 0.0;
	if ((t_1 <= -900.0) || ~((t_1 <= 611.0)))
		tmp = (log(t) * (a - 0.5)) - t;
	else
		tmp = log(((x + y) * z)) + ((log(t) * (a + -0.5)) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -900.0], N[Not[LessEqual[t$95$1, 611.0]], $MachinePrecision]], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -900 or 611 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.8%

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

   3. Taylor expanded in t around -inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. distribute-rgt-neg-in 99.7%

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

      3. mul-1-neg 99.7%

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

      4. unsub-neg 99.7%

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

      5. +-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in t around inf 84.8%

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

      1. neg-mul-1 84.8%

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

   8. Simplified 84.8%

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

   if -900 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 611

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-+l+ 99.4%

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

      4. +-commutative 99.4%

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

      5. fma-define 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. +-commutative 99.4%

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

      2. fma-undefine 99.4%

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

      3. metadata-eval 99.4%

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

      4. sub-neg 99.4%

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

      5. associate-+r+ 99.4%

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

      6. associate--l+ 99.4%

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

      7. +-commutative 99.4%

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

      8. associate-+l- 99.4%

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

      9. sum-log 98.6%

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

      10. sub-neg 98.6%

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

      11. metadata-eval 98.6%

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

      12. *-commutative 98.6%

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

   6. Applied egg-rr 98.6%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -900 \lor \neg \left(\log \left(x + y\right) + \log z \leq 611\right):\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(\log t \cdot \left(a + -0.5\right) - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot \left(a - 0.5\right) - t\\ t_2 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_2 \leq -900 \lor \neg \left(t\_2 \leq 611\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \log \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* (log t) (- a 0.5)) t)) (t_2 (+ (log (+ x y)) (log z))))
   (if (or (<= t_2 -900.0) (not (<= t_2 611.0))) t_1 (+ t_1 (log (* y z))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (log(t) * (a - 0.5)) - t;
	double t_2 = log((x + y)) + log(z);
	double tmp;
	if ((t_2 <= -900.0) || !(t_2 <= 611.0)) {
		tmp = t_1;
	} else {
		tmp = t_1 + log((y * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (log(t) * (a - 0.5d0)) - t
    t_2 = log((x + y)) + log(z)
    if ((t_2 <= (-900.0d0)) .or. (.not. (t_2 <= 611.0d0))) then
        tmp = t_1
    else
        tmp = t_1 + log((y * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (Math.log(t) * (a - 0.5)) - t;
	double t_2 = Math.log((x + y)) + Math.log(z);
	double tmp;
	if ((t_2 <= -900.0) || !(t_2 <= 611.0)) {
		tmp = t_1;
	} else {
		tmp = t_1 + Math.log((y * z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (math.log(t) * (a - 0.5)) - t
	t_2 = math.log((x + y)) + math.log(z)
	tmp = 0
	if (t_2 <= -900.0) or not (t_2 <= 611.0):
		tmp = t_1
	else:
		tmp = t_1 + math.log((y * z))
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(log(t) * Float64(a - 0.5)) - t)
	t_2 = Float64(log(Float64(x + y)) + log(z))
	tmp = 0.0
	if ((t_2 <= -900.0) || !(t_2 <= 611.0))
		tmp = t_1;
	else
		tmp = Float64(t_1 + log(Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (log(t) * (a - 0.5)) - t;
	t_2 = log((x + y)) + log(z);
	tmp = 0.0;
	if ((t_2 <= -900.0) || ~((t_2 <= 611.0)))
		tmp = t_1;
	else
		tmp = t_1 + log((y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -900.0], N[Not[LessEqual[t$95$2, 611.0]], $MachinePrecision]], t$95$1, N[(t$95$1 + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -900 or 611 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.8%

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

   3. Taylor expanded in t around -inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. distribute-rgt-neg-in 99.7%

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

      3. mul-1-neg 99.7%

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

      4. unsub-neg 99.7%

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

      5. +-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in t around inf 84.8%

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

      1. neg-mul-1 84.8%

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

   8. Simplified 84.8%

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

   if -900 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 611

   1. Initial program 99.4%

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

      1. associate-+l- 99.4%

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

      2. associate--l+ 99.4%

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

      3. sub-neg 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. fma-undefine 99.4%

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

      8. sub-neg 99.4%

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

      9. +-commutative 99.4%

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

      10. distribute-neg-in 99.4%

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

      11. metadata-eval 99.4%

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

      12. metadata-eval 99.4%

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

      13. unsub-neg 99.4%

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

   3. Simplified 99.4%

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

      1. associate-+r- 99.4%

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

      2. fma-undefine 99.4%

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

      3. associate--r+ 99.4%

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

      4. sum-log 98.6%

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

   6. Applied egg-rr 98.6%

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

   7. Taylor expanded in x around 0 64.0%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -900 \lor \neg \left(\log \left(x + y\right) + \log z \leq 611\right):\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot \left(a - 0.5\right) - t\right) + \log \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot \left(a - 0.5\right)\\ \mathbf{if}\;t \leq 0.285:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) (- a 0.5))))
   (if (<= t 0.285) (+ (log (+ x y)) (+ (log z) t_1)) (- t_1 t))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * (a - 0.5);
	double tmp;
	if (t <= 0.285) {
		tmp = log((x + y)) + (log(z) + t_1);
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(t) * (a - 0.5d0)
    if (t <= 0.285d0) then
        tmp = log((x + y)) + (log(z) + t_1)
    else
        tmp = t_1 - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log(t) * (a - 0.5);
	double tmp;
	if (t <= 0.285) {
		tmp = Math.log((x + y)) + (Math.log(z) + t_1);
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(t) * (a - 0.5)
	tmp = 0
	if t <= 0.285:
		tmp = math.log((x + y)) + (math.log(z) + t_1)
	else:
		tmp = t_1 - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * Float64(a - 0.5))
	tmp = 0.0
	if (t <= 0.285)
		tmp = Float64(log(Float64(x + y)) + Float64(log(z) + t_1));
	else
		tmp = Float64(t_1 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(t) * (a - 0.5);
	tmp = 0.0;
	if (t <= 0.285)
		tmp = log((x + y)) + (log(z) + t_1);
	else
		tmp = t_1 - 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]}, If[LessEqual[t, 0.285], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 0.284999999999999976

   1. Initial program 99.2%

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

      1. associate-+l- 99.2%

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

      2. associate--l+ 99.1%

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

      3. sub-neg 99.1%

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

      4. +-commutative 99.1%

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

      5. *-commutative 99.1%

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

      6. distribute-rgt-neg-in 99.1%

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

      7. fma-undefine 99.1%

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

      8. sub-neg 99.1%

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

      9. +-commutative 99.1%

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

      10. distribute-neg-in 99.1%

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

      11. metadata-eval 99.1%

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

      12. metadata-eval 99.1%

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

      13. unsub-neg 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in t around 0 97.9%

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

   if 0.284999999999999976 < t

   1. Initial program 99.8%

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

   3. Taylor expanded in t around -inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. distribute-rgt-neg-in 99.9%

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

      3. mul-1-neg 99.9%

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

      4. unsub-neg 99.9%

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

      5. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in t around inf 98.1%

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

      1. neg-mul-1 98.1%

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

   8. Simplified 98.1%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.285:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((log((x + y)) + log(z)) - t) + (log(t) * (a - 0.5));
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[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Final simplification 99.5%

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

Alternative 6: 69.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate--l+ 99.5%

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

   2. +-commutative 99.5%

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

   3. associate-+l+ 99.5%

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

   4. +-commutative 99.5%

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

   5. fma-define 99.5%

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

   6. sub-neg 99.5%

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

   7. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in x around 0 67.8%

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

6. Final simplification 67.8%

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

Alternative 7: 74.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{-42} \lor \neg \left(a \leq 2.1 \cdot 10^{-8}\right):\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot z\right) - \left(t + \log t \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -4.2e-42) (not (<= a 2.1e-8)))
   (- (* (log t) (- a 0.5)) t)
   (- (log (* y z)) (+ t (* (log t) 0.5)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -4.2e-42) || !(a <= 2.1e-8)) {
		tmp = (log(t) * (a - 0.5)) - t;
	} else {
		tmp = log((y * z)) - (t + (log(t) * 0.5));
	}
	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 <= (-4.2d-42)) .or. (.not. (a <= 2.1d-8))) then
        tmp = (log(t) * (a - 0.5d0)) - t
    else
        tmp = log((y * z)) - (t + (log(t) * 0.5d0))
    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 <= -4.2e-42) || !(a <= 2.1e-8)) {
		tmp = (Math.log(t) * (a - 0.5)) - t;
	} else {
		tmp = Math.log((y * z)) - (t + (Math.log(t) * 0.5));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -4.2e-42) or not (a <= 2.1e-8):
		tmp = (math.log(t) * (a - 0.5)) - t
	else:
		tmp = math.log((y * z)) - (t + (math.log(t) * 0.5))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -4.2e-42) || !(a <= 2.1e-8))
		tmp = Float64(Float64(log(t) * Float64(a - 0.5)) - t);
	else
		tmp = Float64(log(Float64(y * z)) - Float64(t + Float64(log(t) * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -4.2e-42) || ~((a <= 2.1e-8)))
		tmp = (log(t) * (a - 0.5)) - t;
	else
		tmp = log((y * z)) - (t + (log(t) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -4.2e-42], N[Not[LessEqual[a, 2.1e-8]], $MachinePrecision]], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] - N[(t + N[(N[Log[t], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.20000000000000013e-42 or 2.09999999999999994e-8 < 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. Add Preprocessing

   3. Taylor expanded in t around -inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. distribute-rgt-neg-in 99.7%

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

      3. mul-1-neg 99.7%

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

      4. unsub-neg 99.7%

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

      5. +-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in t around inf 96.7%

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

      1. neg-mul-1 96.7%

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

   8. Simplified 96.7%

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

   if -4.20000000000000013e-42 < a < 2.09999999999999994e-8

   1. Initial program 99.4%

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

      1. associate-+l- 99.4%

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

      2. associate--l+ 99.4%

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

      3. sub-neg 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. fma-undefine 99.4%

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

      8. sub-neg 99.4%

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

      9. +-commutative 99.4%

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

      10. distribute-neg-in 99.4%

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

      11. metadata-eval 99.4%

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

      12. metadata-eval 99.4%

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

      13. unsub-neg 99.4%

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

   3. Simplified 99.4%

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

      1. associate-+r- 99.4%

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

      2. fma-undefine 99.4%

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

      3. associate--r+ 99.4%

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

      4. sum-log 79.3%

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

   6. Applied egg-rr 79.3%

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

   7. Taylor expanded in x around 0 51.1%

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

      1. *-commutative 51.1%

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

      2. *-commutative 51.1%

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

   9. Simplified 51.1%

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

   10. Taylor expanded in a around 0 50.7%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{-42} \lor \neg \left(a \leq 2.1 \cdot 10^{-8}\right):\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot z\right) - \left(t + \log t \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5.1 \cdot 10^{-154} \lor \neg \left(t \leq 6.5 \cdot 10^{-62}\right):\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot \left(z \cdot \sqrt{\frac{1}{t}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t 5.1e-154) (not (<= t 6.5e-62)))
   (- (* (log t) (- a 0.5)) t)
   (log (* (+ x y) (* z (sqrt (/ 1.0 t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= 5.1e-154) || !(t <= 6.5e-62)) {
		tmp = (log(t) * (a - 0.5)) - t;
	} else {
		tmp = log(((x + y) * (z * sqrt((1.0 / t)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((t <= 5.1d-154) .or. (.not. (t <= 6.5d-62))) then
        tmp = (log(t) * (a - 0.5d0)) - t
    else
        tmp = log(((x + y) * (z * sqrt((1.0d0 / t)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= 5.1e-154) || !(t <= 6.5e-62)) {
		tmp = (Math.log(t) * (a - 0.5)) - t;
	} else {
		tmp = Math.log(((x + y) * (z * Math.sqrt((1.0 / t)))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= 5.1e-154) or not (t <= 6.5e-62):
		tmp = (math.log(t) * (a - 0.5)) - t
	else:
		tmp = math.log(((x + y) * (z * math.sqrt((1.0 / t)))))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= 5.1e-154) || !(t <= 6.5e-62))
		tmp = Float64(Float64(log(t) * Float64(a - 0.5)) - t);
	else
		tmp = log(Float64(Float64(x + y) * Float64(z * sqrt(Float64(1.0 / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= 5.1e-154) || ~((t <= 6.5e-62)))
		tmp = (log(t) * (a - 0.5)) - t;
	else
		tmp = log(((x + y) * (z * sqrt((1.0 / t)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, 5.1e-154], N[Not[LessEqual[t, 6.5e-62]], $MachinePrecision]], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[Log[N[(N[(x + y), $MachinePrecision] * N[(z * N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.0999999999999998e-154 or 6.50000000000000026e-62 < t

   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. Add Preprocessing

   3. Taylor expanded in t around -inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. distribute-rgt-neg-in 99.6%

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

      3. mul-1-neg 99.6%

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

      4. unsub-neg 99.6%

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

      5. +-commutative 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in t around inf 80.3%

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

      1. neg-mul-1 80.3%

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

   8. Simplified 80.3%

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

   if 5.0999999999999998e-154 < t < 6.50000000000000026e-62

   1. Initial program 99.0%

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

      1. associate--l+ 99.0%

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

      2. +-commutative 99.0%

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

      3. associate-+l+ 98.8%

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

      4. +-commutative 98.8%

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

      5. fma-define 98.9%

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

      6. sub-neg 98.9%

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

      7. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

      1. add-log-exp 66.1%

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

      2. +-commutative 66.1%

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

      3. exp-sum 60.7%

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

      4. fma-undefine 60.6%

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

      5. metadata-eval 60.6%

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

      6. sub-neg 60.6%

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

      7. exp-sum 60.8%

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

      8. add-exp-log 61.0%

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

      9. sub-neg 61.0%

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

      10. metadata-eval 61.0%

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

      11. *-commutative 61.0%

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

      12. exp-to-pow 61.1%

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

      13. exp-diff 61.1%

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

      14. add-exp-log 61.6%

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

   6. Applied egg-rr 61.6%

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

      1. associate-*l* 64.1%

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

      2. associate-*r/ 64.1%

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

      3. *-commutative 64.1%

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

      4. +-commutative 64.1%

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

   8. Simplified 64.1%

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

   9. Taylor expanded in t around 0 61.2%

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

       1. *-commutative 61.2%

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

       2. exp-to-pow 61.6%

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

       3. sub-neg 61.6%

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

       4. metadata-eval 61.6%

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

       5. +-commutative 61.6%

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

   11. Simplified 61.6%

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

   12. Taylor expanded in a around 0 63.1%

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

       1. associate-*r* 65.8%

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

   14. Simplified 65.8%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.1 \cdot 10^{-154} \lor \neg \left(t \leq 6.5 \cdot 10^{-62}\right):\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot \left(z \cdot \sqrt{\frac{1}{t}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5.1 \cdot 10^{-154} \lor \neg \left(t \leq 1.8 \cdot 10^{-87}\right):\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(y \cdot z\right) \cdot {t}^{\left(a + -0.5\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t 5.1e-154) (not (<= t 1.8e-87)))
   (- (* (log t) (- a 0.5)) t)
   (log (* (* y z) (pow t (+ a -0.5))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= 5.1e-154) || !(t <= 1.8e-87)) {
		tmp = (log(t) * (a - 0.5)) - t;
	} else {
		tmp = log(((y * z) * pow(t, (a + -0.5))));
	}
	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 <= 5.1d-154) .or. (.not. (t <= 1.8d-87))) then
        tmp = (log(t) * (a - 0.5d0)) - t
    else
        tmp = log(((y * z) * (t ** (a + (-0.5d0)))))
    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 <= 5.1e-154) || !(t <= 1.8e-87)) {
		tmp = (Math.log(t) * (a - 0.5)) - t;
	} else {
		tmp = Math.log(((y * z) * Math.pow(t, (a + -0.5))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= 5.1e-154) or not (t <= 1.8e-87):
		tmp = (math.log(t) * (a - 0.5)) - t
	else:
		tmp = math.log(((y * z) * math.pow(t, (a + -0.5))))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= 5.1e-154) || !(t <= 1.8e-87))
		tmp = Float64(Float64(log(t) * Float64(a - 0.5)) - t);
	else
		tmp = log(Float64(Float64(y * z) * (t ^ Float64(a + -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= 5.1e-154) || ~((t <= 1.8e-87)))
		tmp = (log(t) * (a - 0.5)) - t;
	else
		tmp = log(((y * z) * (t ^ (a + -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, 5.1e-154], N[Not[LessEqual[t, 1.8e-87]], $MachinePrecision]], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[Log[N[(N[(y * z), $MachinePrecision] * N[Power[t, N[(a + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.0999999999999998e-154 or 1.79999999999999996e-87 < t

   1. Initial program 99.5%

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

   3. Taylor expanded in t around -inf 99.5%

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

      1. mul-1-neg 99.5%

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

      2. distribute-rgt-neg-in 99.5%

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

      3. mul-1-neg 99.5%

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

      4. unsub-neg 99.5%

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

      5. +-commutative 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in t around inf 78.8%

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

      1. neg-mul-1 78.8%

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

   8. Simplified 78.8%

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

   if 5.0999999999999998e-154 < t < 1.79999999999999996e-87

   1. Initial program 99.2%

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

      1. associate--l+ 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-+l+ 98.9%

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

      4. +-commutative 98.9%

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

      5. fma-define 99.0%

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

      6. sub-neg 99.0%

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

      7. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

      1. add-log-exp 73.4%

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

      2. +-commutative 73.4%

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

      3. exp-sum 66.2%

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

      4. fma-undefine 66.1%

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

      5. metadata-eval 66.1%

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

      6. sub-neg 66.1%

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

      7. exp-sum 66.3%

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

      8. add-exp-log 66.4%

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

      9. sub-neg 66.4%

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

      10. metadata-eval 66.4%

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

      11. *-commutative 66.4%

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

      12. exp-to-pow 66.6%

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

      13. exp-diff 66.6%

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

      14. add-exp-log 66.9%

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

   6. Applied egg-rr 66.9%

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

      1. associate-*l* 73.9%

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

      2. associate-*r/ 73.9%

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

      3. *-commutative 73.9%

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

      4. +-commutative 73.9%

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

   8. Simplified 73.9%

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

   9. Taylor expanded in t around 0 66.6%

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

       1. *-commutative 66.6%

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

       2. exp-to-pow 66.9%

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

       3. sub-neg 66.9%

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

       4. metadata-eval 66.9%

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

       5. +-commutative 66.9%

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

   11. Simplified 66.9%

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

   12. Taylor expanded in x around 0 36.3%

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

       1. associate-*r* 36.2%

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

       2. *-commutative 36.2%

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

       3. exp-to-pow 36.3%

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

       4. sub-neg 36.3%

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

       5. metadata-eval 36.3%

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

   14. Simplified 36.3%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.1 \cdot 10^{-154} \lor \neg \left(t \leq 1.8 \cdot 10^{-87}\right):\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(y \cdot z\right) \cdot {t}^{\left(a + -0.5\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4.3 \cdot 10^{-154} \lor \neg \left(t \leq 1.4 \cdot 10^{-61}\right):\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(z \cdot \left(y \cdot {t}^{\left(a + -0.5\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t 4.3e-154) (not (<= t 1.4e-61)))
   (- (* (log t) (- a 0.5)) t)
   (log (* z (* y (pow t (+ a -0.5)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= 4.3e-154) || !(t <= 1.4e-61)) {
		tmp = (log(t) * (a - 0.5)) - t;
	} else {
		tmp = log((z * (y * pow(t, (a + -0.5)))));
	}
	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 <= 4.3d-154) .or. (.not. (t <= 1.4d-61))) then
        tmp = (log(t) * (a - 0.5d0)) - t
    else
        tmp = log((z * (y * (t ** (a + (-0.5d0))))))
    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 <= 4.3e-154) || !(t <= 1.4e-61)) {
		tmp = (Math.log(t) * (a - 0.5)) - t;
	} else {
		tmp = Math.log((z * (y * Math.pow(t, (a + -0.5)))));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= 4.3e-154) or not (t <= 1.4e-61):
		tmp = (math.log(t) * (a - 0.5)) - t
	else:
		tmp = math.log((z * (y * math.pow(t, (a + -0.5)))))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= 4.3e-154) || !(t <= 1.4e-61))
		tmp = Float64(Float64(log(t) * Float64(a - 0.5)) - t);
	else
		tmp = log(Float64(z * Float64(y * (t ^ Float64(a + -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= 4.3e-154) || ~((t <= 1.4e-61)))
		tmp = (log(t) * (a - 0.5)) - t;
	else
		tmp = log((z * (y * (t ^ (a + -0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, 4.3e-154], N[Not[LessEqual[t, 1.4e-61]], $MachinePrecision]], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[Log[N[(z * N[(y * N[Power[t, N[(a + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.29999999999999992e-154 or 1.4000000000000001e-61 < t

   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. Add Preprocessing

   3. Taylor expanded in t around -inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. distribute-rgt-neg-in 99.6%

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

      3. mul-1-neg 99.6%

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

      4. unsub-neg 99.6%

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

      5. +-commutative 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in t around inf 80.3%

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

      1. neg-mul-1 80.3%

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

   8. Simplified 80.3%

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

   if 4.29999999999999992e-154 < t < 1.4000000000000001e-61

   1. Initial program 99.0%

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

      1. associate--l+ 99.0%

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

      2. +-commutative 99.0%

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

      3. associate-+l+ 98.8%

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

      4. +-commutative 98.8%

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

      5. fma-define 98.9%

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

      6. sub-neg 98.9%

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

      7. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

      1. add-log-exp 66.1%

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

      2. +-commutative 66.1%

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

      3. exp-sum 60.7%

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

      4. fma-undefine 60.6%

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

      5. metadata-eval 60.6%

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

      6. sub-neg 60.6%

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

      7. exp-sum 60.8%

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

      8. add-exp-log 61.0%

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

      9. sub-neg 61.0%

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

      10. metadata-eval 61.0%

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

      11. *-commutative 61.0%

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

      12. exp-to-pow 61.1%

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

      13. exp-diff 61.1%

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

      14. add-exp-log 61.6%

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

   6. Applied egg-rr 61.6%

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

      1. associate-*l* 64.1%

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

      2. associate-*r/ 64.1%

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

      3. *-commutative 64.1%

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

      4. +-commutative 64.1%

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

   8. Simplified 64.1%

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

   9. Taylor expanded in t around 0 61.2%

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

       1. *-commutative 61.2%

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

       2. exp-to-pow 61.6%

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

       3. sub-neg 61.6%

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

       4. metadata-eval 61.6%

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

       5. +-commutative 61.6%

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

   11. Simplified 61.6%

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

   12. Taylor expanded in x around 0 33.9%

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

       1. exp-to-pow 34.2%

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

       2. sub-neg 34.2%

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

       3. metadata-eval 34.2%

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

   14. Simplified 34.2%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.3 \cdot 10^{-154} \lor \neg \left(t \leq 1.4 \cdot 10^{-61}\right):\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(z \cdot \left(y \cdot {t}^{\left(a + -0.5\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 8500000000000:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 8500000000000.0) (* (log t) a) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 8500000000000.0) {
		tmp = log(t) * a;
	} 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 <= 8500000000000.0d0) then
        tmp = log(t) * a
    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 <= 8500000000000.0) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 8500000000000.0:
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 8500000000000.0)
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 8500000000000.0)
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 8500000000000.0], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 8.5e12

   1. Initial program 99.2%

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

      1. associate-+l- 99.2%

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

      2. associate--l+ 99.1%

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

      3. sub-neg 99.1%

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

      4. +-commutative 99.1%

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

      5. *-commutative 99.1%

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

      6. distribute-rgt-neg-in 99.1%

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

      7. fma-undefine 99.1%

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

      8. sub-neg 99.1%

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

      9. +-commutative 99.1%

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

      10. distribute-neg-in 99.1%

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

      11. metadata-eval 99.1%

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

      12. metadata-eval 99.1%

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

      13. unsub-neg 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in a around inf 43.7%

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

      1. *-commutative 43.7%

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

   7. Simplified 43.7%

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

   if 8.5e12 < 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 80.5%

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

      1. neg-mul-1 80.5%

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

   7. Simplified 80.5%

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

Alternative 12: 76.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in t around -inf 99.5%

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

   1. mul-1-neg 99.5%

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

   2. distribute-rgt-neg-in 99.5%

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

   3. mul-1-neg 99.5%

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

   4. unsub-neg 99.5%

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

   5. +-commutative 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in t around inf 73.8%

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

   1. neg-mul-1 73.8%

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

8. Simplified 73.8%

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

9. Final simplification 73.8%

   \[\leadsto \log t \cdot \left(a - 0.5\right) - t \]
10. Add Preprocessing

Alternative 13: 37.4% accurate, 156.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-+l- 99.5%

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

   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 40.6%

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

   1. neg-mul-1 40.6%

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

7. Simplified 40.6%

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

Developer Target 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024186 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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