Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 20.3s

Alternatives: 14

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

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

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+ 99.6%

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

   3. sub-neg 99.6%

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

   4. +-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. distribute-rgt-neg-in 99.6%

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

   7. fma-undefine 99.6%

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

   8. sub-neg 99.6%

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

   9. +-commutative 99.6%

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

   10. distribute-neg-in 99.6%

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

   11. metadata-eval 99.6%

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

   12. metadata-eval 99.6%

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

   13. unsub-neg 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 63.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq -800:\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{elif}\;t\_1 \leq 705:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \left(0.5 - a\right) \cdot \log \left(\frac{1}{t}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (<= t_1 -800.0)
     (- (+ (log y) (* (log t) a)) t)
     (if (<= t_1 705.0)
       (- (+ (log (* y z)) (* (- 0.5 a) (log (/ 1.0 t)))) t)
       (+ (log y) (fma (log t) a (- 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 <= -800.0) {
		tmp = (log(y) + (log(t) * a)) - t;
	} else if (t_1 <= 705.0) {
		tmp = (log((y * z)) + ((0.5 - a) * log((1.0 / t)))) - t;
	} else {
		tmp = log(y) + fma(log(t), a, -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 <= -800.0)
		tmp = Float64(Float64(log(y) + Float64(log(t) * a)) - t);
	elseif (t_1 <= 705.0)
		tmp = Float64(Float64(log(Float64(y * z)) + Float64(Float64(0.5 - a) * log(Float64(1.0 / t)))) - t);
	else
		tmp = Float64(log(y) + fma(log(t), a, Float64(-t)));
	end
	return 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[LessEqual[t$95$1, -800.0], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$1, 705.0], N[(N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 - a), $MachinePrecision] * N[Log[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a + (-t)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -800

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 66.7%

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

   6. Taylor expanded in a around inf 66.7%

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

      1. *-commutative 66.7%

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

   8. Simplified 66.7%

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

   if -800 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 705

   1. Initial program 99.5%

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

      1. associate-+l- 99.5%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-undefine 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

      1. associate-+r- 99.6%

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

      2. fma-undefine 99.5%

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

      3. associate--r+ 99.5%

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

      4. +-commutative 99.5%

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

      5. sum-log 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in t around inf 99.6%

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

   8. Taylor expanded in x around 0 59.6%

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

   if 705 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   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+ 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 81.8%

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

   6. Taylor expanded in a around inf 65.9%

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

      1. *-commutative 65.9%

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

   8. Simplified 65.9%

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

      1. associate--l+ 65.9%

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

      2. fma-neg 65.9%

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

   10. Applied egg-rr 65.9%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -800:\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{elif}\;\log \left(x + y\right) + \log z \leq 705:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \left(0.5 - a\right) \cdot \log \left(\frac{1}{t}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq -800 \lor \neg \left(t\_1 \leq 705\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \left(0.5 - a\right) \cdot \log \left(\frac{1}{t}\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (or (<= t_1 -800.0) (not (<= t_1 705.0)))
     (- (+ (log y) (* (log t) a)) t)
     (- (+ (log (* y z)) (* (- 0.5 a) (log (/ 1.0 t)))) 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 <= -800.0) || !(t_1 <= 705.0)) {
		tmp = (log(y) + (log(t) * a)) - t;
	} else {
		tmp = (log((y * z)) + ((0.5 - a) * log((1.0 / t)))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log((x + y)) + log(z)
    if ((t_1 <= (-800.0d0)) .or. (.not. (t_1 <= 705.0d0))) then
        tmp = (log(y) + (log(t) * a)) - t
    else
        tmp = (log((y * z)) + ((0.5d0 - a) * log((1.0d0 / t)))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log((x + y)) + Math.log(z);
	double tmp;
	if ((t_1 <= -800.0) || !(t_1 <= 705.0)) {
		tmp = (Math.log(y) + (Math.log(t) * a)) - t;
	} else {
		tmp = (Math.log((y * z)) + ((0.5 - a) * Math.log((1.0 / t)))) - 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 <= -800.0) or not (t_1 <= 705.0):
		tmp = (math.log(y) + (math.log(t) * a)) - t
	else:
		tmp = (math.log((y * z)) + ((0.5 - a) * math.log((1.0 / t)))) - 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 <= -800.0) || !(t_1 <= 705.0))
		tmp = Float64(Float64(log(y) + Float64(log(t) * a)) - t);
	else
		tmp = Float64(Float64(log(Float64(y * z)) + Float64(Float64(0.5 - a) * log(Float64(1.0 / t)))) - 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 <= -800.0) || ~((t_1 <= 705.0)))
		tmp = (log(y) + (log(t) * a)) - t;
	else
		tmp = (log((y * z)) + ((0.5 - a) * log((1.0 / t)))) - 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, -800.0], N[Not[LessEqual[t$95$1, 705.0]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 - a), $MachinePrecision] * N[Log[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -800 or 705 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 80.3%

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

   6. Taylor expanded in a around inf 66.0%

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

      1. *-commutative 66.0%

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

   8. Simplified 66.0%

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

   if -800 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 705

   1. Initial program 99.5%

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

      1. associate-+l- 99.5%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-undefine 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

      1. associate-+r- 99.6%

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

      2. fma-undefine 99.5%

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

      3. associate--r+ 99.5%

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

      4. +-commutative 99.5%

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

      5. sum-log 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in t around inf 99.6%

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

   8. Taylor expanded in x around 0 59.6%

      \[\leadsto \color{blue}{\left(\log \left(y \cdot z\right) - -1 \cdot \left(\log \left(\frac{1}{t}\right) \cdot \left(0.5 - a\right)\right)\right)} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -800 \lor \neg \left(\log \left(x + y\right) + \log z \leq 705\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \left(0.5 - a\right) \cdot \log \left(\frac{1}{t}\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq -800 \lor \neg \left(t\_1 \leq 705\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (or (<= t_1 -800.0) (not (<= t_1 705.0)))
     (- (+ (log y) (* (log t) a)) t)
     (- (+ (log (* 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 <= -800.0) || !(t_1 <= 705.0)) {
		tmp = (log(y) + (log(t) * a)) - t;
	} else {
		tmp = (log((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 <= (-800.0d0)) .or. (.not. (t_1 <= 705.0d0))) then
        tmp = (log(y) + (log(t) * a)) - t
    else
        tmp = (log((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 <= -800.0) || !(t_1 <= 705.0)) {
		tmp = (Math.log(y) + (Math.log(t) * a)) - t;
	} else {
		tmp = (Math.log((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 <= -800.0) or not (t_1 <= 705.0):
		tmp = (math.log(y) + (math.log(t) * a)) - t
	else:
		tmp = (math.log((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 <= -800.0) || !(t_1 <= 705.0))
		tmp = Float64(Float64(log(y) + Float64(log(t) * a)) - t);
	else
		tmp = Float64(Float64(log(Float64(y * z)) + 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 <= -800.0) || ~((t_1 <= 705.0)))
		tmp = (log(y) + (log(t) * a)) - t;
	else
		tmp = (log((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, -800.0], N[Not[LessEqual[t$95$1, 705.0]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -800 or 705 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 80.3%

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

   6. Taylor expanded in a around inf 66.0%

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

      1. *-commutative 66.0%

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

   8. Simplified 66.0%

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

   if -800 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 705

   1. Initial program 99.5%

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

      1. associate-+l- 99.5%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-undefine 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

      1. associate-+r- 99.6%

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

      2. fma-undefine 99.5%

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

      3. associate--r+ 99.5%

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

      4. +-commutative 99.5%

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

      5. sum-log 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around 0 59.6%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -800 \lor \neg \left(\log \left(x + y\right) + \log z \leq 705\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-define 99.6%

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

   3. sub-neg 99.6%

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

   4. metadata-eval 99.6%

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

   5. associate--l+ 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 68.3%

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

   1. +-commutative 68.3%

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

7. Simplified 68.3%

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

8. Final simplification 68.3%

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

Alternative 6: 68.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -100 \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log z + \log y\right) - t\right) + \log t \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -100.0) (not (<= (- a 0.5) -0.4)))
   (+ (log y) (fma (log t) a (- t)))
   (+ (- (+ (log z) (log y)) t) (* (log t) -0.5))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -100.0) || !((a - 0.5) <= -0.4)) {
		tmp = log(y) + fma(log(t), a, -t);
	} else {
		tmp = ((log(z) + log(y)) - t) + (log(t) * -0.5);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -100.0) || !(Float64(a - 0.5) <= -0.4))
		tmp = Float64(log(y) + fma(log(t), a, Float64(-t)));
	else
		tmp = Float64(Float64(Float64(log(z) + log(y)) - t) + Float64(log(t) * -0.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -100.0], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a + (-t)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a 1/2) < -100 or -0.40000000000000002 < (-.f64 a 1/2)

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 67.9%

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

   6. Taylor expanded in a around inf 67.0%

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

      1. *-commutative 67.0%

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

   8. Simplified 67.0%

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

      1. associate--l+ 67.0%

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

      2. fma-neg 67.0%

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

   10. Applied egg-rr 67.0%

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

   if -100 < (-.f64 a 1/2) < -0.40000000000000002

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. sub-neg 99.5%

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

      3. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. metadata-eval 99.5%

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

      2. sub-neg 99.5%

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

      3. flip-- 99.5%

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

      4. metadata-eval 99.5%

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

      5. metadata-eval 99.5%

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

      6. associate-*l/ 99.5%

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

      7. fma-neg 99.5%

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

      8. metadata-eval 99.5%

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

      9. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around 0 68.8%

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

      1. +-commutative 68.8%

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

   9. Simplified 68.8%

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

   10. Taylor expanded in a around 0 68.5%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -100 \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log z + \log y\right) - t\right) + \log t \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[Log[z], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision]), $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. Step-by-step derivation

   1. associate--l+ 99.5%

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

   2. sub-neg 99.5%

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

   3. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in x around 0 68.3%

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

   1. remove-double-neg 68.3%

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

   2. log-rec 68.3%

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

   3. mul-1-neg 68.3%

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

   4. +-commutative 68.3%

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

   5. associate--l+ 68.3%

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

   6. mul-1-neg 68.3%

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

   7. log-rec 68.3%

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

   8. remove-double-neg 68.3%

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

7. Simplified 68.3%

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

8. Final simplification 68.3%

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

Alternative 8: 61.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.05e-5)
   (+ (* (log t) (+ a -0.5)) (log (* y z)))
   (- (+ (log y) (* (log t) a)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.05e-5) {
		tmp = (log(t) * (a + -0.5)) + log((y * z));
	} else {
		tmp = (log(y) + (log(t) * a)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 1.05d-5) then
        tmp = (log(t) * (a + (-0.5d0))) + log((y * z))
    else
        tmp = (log(y) + (log(t) * a)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.05e-5) {
		tmp = (Math.log(t) * (a + -0.5)) + Math.log((y * z));
	} else {
		tmp = (Math.log(y) + (Math.log(t) * a)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.05e-5:
		tmp = (math.log(t) * (a + -0.5)) + math.log((y * z))
	else:
		tmp = (math.log(y) + (math.log(t) * a)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.05e-5)
		tmp = Float64(Float64(log(t) * Float64(a + -0.5)) + log(Float64(y * z)));
	else
		tmp = Float64(Float64(log(y) + Float64(log(t) * a)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 1.05e-5)
		tmp = (log(t) * (a + -0.5)) + log((y * z));
	else
		tmp = (log(y) + (log(t) * a)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.05e-5], N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.04999999999999994e-5

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. fma-define 99.3%

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

      3. sub-neg 99.3%

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

      4. metadata-eval 99.3%

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

      5. associate--l+ 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 59.4%

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

      1. +-commutative 59.4%

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

   7. Simplified 59.4%

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

   8. Taylor expanded in t around 0 58.8%

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

      1. associate-+r+ 58.8%

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

      2. log-prod 39.7%

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

      3. sub-neg 39.7%

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

      4. metadata-eval 39.7%

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

      5. +-commutative 39.7%

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

   10. Simplified 39.7%

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

   if 1.04999999999999994e-5 < t

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 76.5%

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

   6. Taylor expanded in a around inf 73.6%

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

      1. *-commutative 73.6%

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

   8. Simplified 73.6%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{+48}:\\ \;\;\;\;\log y + \log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2.8e+48) (+ (log y) (* (log t) a)) (+ (log (+ x y)) (- (log z) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.8e+48) {
		tmp = log(y) + (log(t) * a);
	} else {
		tmp = log((x + y)) + (log(z) - t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 2.8d+48) then
        tmp = log(y) + (log(t) * a)
    else
        tmp = log((x + y)) + (log(z) - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.8e+48) {
		tmp = Math.log(y) + (Math.log(t) * a);
	} else {
		tmp = Math.log((x + y)) + (Math.log(z) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 2.8e+48:
		tmp = math.log(y) + (math.log(t) * a)
	else:
		tmp = math.log((x + y)) + (math.log(z) - t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.8e+48)
		tmp = Float64(log(y) + Float64(log(t) * a));
	else
		tmp = Float64(log(Float64(x + y)) + Float64(log(z) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 2.8e+48)
		tmp = log(y) + (log(t) * a);
	else
		tmp = log((x + y)) + (log(z) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.8e+48], N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.80000000000000012e48

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. sub-neg 99.4%

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

      3. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 61.3%

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

   6. Taylor expanded in a around inf 39.6%

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

      1. *-commutative 39.6%

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

   8. Simplified 39.6%

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

   9. Taylor expanded in t around 0 37.7%

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

   if 2.80000000000000012e48 < t

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

      2. associate--l+ 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      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 81.6%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.8 \cdot 10^{+48}:\\ \;\;\;\;\log y + \log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.45 \cdot 10^{+51}:\\ \;\;\;\;\log y + \log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2.45e+51) (+ (log y) (* (log t) a)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.45e+51) {
		tmp = log(y) + (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 <= 2.45d+51) then
        tmp = log(y) + (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 <= 2.45e+51) {
		tmp = Math.log(y) + (Math.log(t) * a);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 2.45e+51:
		tmp = math.log(y) + (math.log(t) * a)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.45e+51)
		tmp = Float64(log(y) + 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 <= 2.45e+51)
		tmp = log(y) + (log(t) * a);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.45e+51], N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 2.44999999999999992e51

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. sub-neg 99.4%

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

      3. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 61.3%

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

   6. Taylor expanded in a around inf 39.6%

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

      1. *-commutative 39.6%

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

   8. Simplified 39.6%

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

   9. Taylor expanded in t around 0 37.7%

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

   if 2.44999999999999992e51 < t

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. fma-define 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate--l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 77.9%

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

      1. +-commutative 77.9%

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

   7. Simplified 77.9%

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

   8. Taylor expanded in t around inf 81.6%

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

      1. neg-mul-1 81.6%

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

   10. Simplified 81.6%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.45 \cdot 10^{+51}:\\ \;\;\;\;\log y + \log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $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. Step-by-step derivation

   1. associate--l+ 99.5%

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

   2. sub-neg 99.5%

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

   3. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in x around 0 68.3%

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

6. Taylor expanded in a around inf 55.9%

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

   1. *-commutative 55.9%

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

8. Simplified 55.9%

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

9. Final simplification 55.9%

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

Alternative 12: 41.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 370:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (if (<= t 370.0) (log (+ x y)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 370.0) {
		tmp = log((x + y));
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 370.0d0) then
        tmp = log((x + y))
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 370.0) {
		tmp = Math.log((x + y));
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 370.0:
		tmp = math.log((x + y))
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 370.0)
		tmp = log(Float64(x + y));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 370.0)
		tmp = log((x + y));
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 370.0], N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 370

   1. Initial program 99.3%

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

      1. associate-+l- 99.3%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. fma-undefine 99.3%

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

      8. sub-neg 99.3%

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

      9. +-commutative 99.3%

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

      10. distribute-neg-in 99.3%

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

      11. metadata-eval 99.3%

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

      12. metadata-eval 99.3%

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

      13. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in t around inf 8.9%

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

      1. neg-mul-1 8.9%

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

   7. Simplified 8.9%

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

   8. Taylor expanded in t around 0 8.9%

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

      1. +-commutative 8.9%

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

   10. Simplified 8.9%

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

   if 370 < t

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. fma-define 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate--l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 75.5%

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

      1. +-commutative 75.5%

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

   7. Simplified 75.5%

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

   8. Taylor expanded in t around inf 75.6%

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

      1. neg-mul-1 75.6%

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

   10. Simplified 75.6%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 370:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{+48}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.9e+48) (* (log t) a) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.9e+48) {
		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 <= 1.9d+48) 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 <= 1.9e+48) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.9e+48:
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.9e+48)
		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 <= 1.9e+48)
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.9e+48], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 1.9e48

   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. +-commutative 99.4%

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

      2. fma-define 99.4%

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

      3. sub-neg 99.4%

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

      4. metadata-eval 99.4%

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

      5. associate--l+ 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 61.3%

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

      1. +-commutative 61.3%

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

   7. Simplified 61.3%

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

   8. Taylor expanded in a around inf 50.3%

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

      1. *-commutative 50.3%

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

   10. Simplified 50.3%

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

   if 1.9e48 < t

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. fma-define 99.9%

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

      3. sub-neg 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate--l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 77.9%

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

      1. +-commutative 77.9%

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

   7. Simplified 77.9%

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

   8. Taylor expanded in t around inf 81.6%

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

      1. neg-mul-1 81.6%

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

   10. Simplified 81.6%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{+48}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 38.4% 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. +-commutative 99.5%

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

   2. fma-define 99.6%

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

   3. sub-neg 99.6%

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

   4. metadata-eval 99.6%

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

   5. associate--l+ 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 68.3%

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

   1. +-commutative 68.3%

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

7. Simplified 68.3%

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

8. Taylor expanded in t around inf 38.9%

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

   1. neg-mul-1 38.9%

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

10. Simplified 38.9%

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

11. Final simplification 38.9%

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

Developer target: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024043 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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