Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 30.5s

Alternatives: 19

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. associate-+l+ 99.6%

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

   3. +-commutative 99.6%

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

   4. associate-+r- 99.6%

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

   5. fma-def 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 65.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log z \leq 295:\\ \;\;\;\;\log \left(y \cdot z\right) + \left(\left(a + -0.5\right) \cdot \log t - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (log z) 295.0)
   (+ (log (* y z)) (- (* (+ a -0.5) (log t)) t))
   (+ (- (log z) t) (* a (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (log(z) <= 295.0) {
		tmp = log((y * z)) + (((a + -0.5) * log(t)) - t);
	} else {
		tmp = (log(z) - t) + (a * log(t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (log(z) <= 295.0d0) then
        tmp = log((y * z)) + (((a + (-0.5d0)) * log(t)) - t)
    else
        tmp = (log(z) - t) + (a * log(t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (Math.log(z) <= 295.0) {
		tmp = Math.log((y * z)) + (((a + -0.5) * Math.log(t)) - t);
	} else {
		tmp = (Math.log(z) - t) + (a * Math.log(t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if math.log(z) <= 295.0:
		tmp = math.log((y * z)) + (((a + -0.5) * math.log(t)) - t)
	else:
		tmp = (math.log(z) - t) + (a * math.log(t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (log(z) <= 295.0)
		tmp = Float64(log(Float64(y * z)) + Float64(Float64(Float64(a + -0.5) * log(t)) - t));
	else
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (log(z) <= 295.0)
		tmp = log((y * z)) + (((a + -0.5) * log(t)) - t);
	else
		tmp = (log(z) - t) + (a * log(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[Log[z], $MachinePrecision], 295.0], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 z) < 295

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.5%

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

      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-def 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. Taylor expanded in x around 0 71.4%

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

   5. Taylor expanded in z around 0 71.4%

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

      1. associate-+r+ 71.4%

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

      2. log-prod 58.9%

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

      3. associate--l+ 58.9%

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

      4. sub-neg 58.9%

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

      5. metadata-eval 58.9%

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

      6. +-commutative 58.9%

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

      7. distribute-rgt-out 58.9%

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

      8. +-commutative 58.9%

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

      9. distribute-rgt-in 58.9%

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

   7. Simplified 58.9%

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

   if 295 < (log.f64 z)

   1. Initial program 99.8%

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

      1. associate--l+ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-def 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 62.9%

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

   5. Taylor expanded in a around inf 74.7%

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

      1. *-commutative 74.7%

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

   7. Simplified 74.7%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z \leq 295:\\ \;\;\;\;\log \left(y \cdot z\right) + \left(\left(a + -0.5\right) \cdot \log t - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. cancel-sign-sub 99.6%

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

   2. cancel-sign-sub-inv 99.6%

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

   3. associate--l+ 99.6%

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

   4. remove-double-neg 99.6%

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

   5. sub-neg 99.6%

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

   6. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 4: 64.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.2 \cdot 10^{-15} \lor \neg \left(a \leq 0.2\right):\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -7.2e-15) (not (<= a 0.2)))
   (- (+ (log y) (* a (log t))) t)
   (- (+ (log y) (log (* z (pow t -0.5)))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -7.2e-15) || !(a <= 0.2)) {
		tmp = (log(y) + (a * log(t))) - t;
	} else {
		tmp = (log(y) + log((z * pow(t, -0.5)))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-7.2d-15)) .or. (.not. (a <= 0.2d0))) then
        tmp = (log(y) + (a * log(t))) - t
    else
        tmp = (log(y) + log((z * (t ** (-0.5d0))))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -7.2e-15) || !(a <= 0.2)) {
		tmp = (Math.log(y) + (a * Math.log(t))) - t;
	} else {
		tmp = (Math.log(y) + Math.log((z * Math.pow(t, -0.5)))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -7.2e-15) or not (a <= 0.2):
		tmp = (math.log(y) + (a * math.log(t))) - t
	else:
		tmp = (math.log(y) + math.log((z * math.pow(t, -0.5)))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -7.2e-15) || !(a <= 0.2))
		tmp = Float64(Float64(log(y) + Float64(a * log(t))) - t);
	else
		tmp = Float64(Float64(log(y) + log(Float64(z * (t ^ -0.5)))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -7.2e-15) || ~((a <= 0.2)))
		tmp = (log(y) + (a * log(t))) - t;
	else
		tmp = (log(y) + log((z * (t ^ -0.5)))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -7.2e-15], N[Not[LessEqual[a, 0.2]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] + N[Log[N[(z * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.2000000000000002e-15 or 0.20000000000000001 < a

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r- 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 96.7%

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

      1. *-commutative 96.7%

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

   6. Simplified 96.7%

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

   7. Taylor expanded in x around 0 75.7%

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

   if -7.2000000000000002e-15 < a < 0.20000000000000001

   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-def 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. Taylor expanded in x around 0 61.1%

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

   5. Taylor expanded in a around 0 60.6%

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

   6. Taylor expanded in z around inf 60.6%

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

      1. *-commutative 60.6%

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

      2. mul-1-neg 60.6%

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

      3. log-rec 60.6%

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

      4. remove-double-neg 60.6%

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

      5. rem-log-exp 51.7%

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

      6. exp-sum 51.7%

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

      7. rem-exp-log 51.7%

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

      8. exp-to-pow 51.7%

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

   8. Simplified 51.7%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.2 \cdot 10^{-15} \lor \neg \left(a \leq 0.2\right):\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \]

Alternative 5: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 120:\\ \;\;\;\;\log z + \left(\log y + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \left(a \cdot \log t - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 120.0)
   (+ (log z) (+ (log y) (* (log t) (- a 0.5))))
   (+ (log (+ x y)) (- (* a (log t)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 120.0) {
		tmp = log(z) + (log(y) + (log(t) * (a - 0.5)));
	} else {
		tmp = log((x + y)) + ((a * log(t)) - t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 120.0d0) then
        tmp = log(z) + (log(y) + (log(t) * (a - 0.5d0)))
    else
        tmp = log((x + y)) + ((a * log(t)) - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 120.0) {
		tmp = Math.log(z) + (Math.log(y) + (Math.log(t) * (a - 0.5)));
	} else {
		tmp = Math.log((x + y)) + ((a * Math.log(t)) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 120.0:
		tmp = math.log(z) + (math.log(y) + (math.log(t) * (a - 0.5)))
	else:
		tmp = math.log((x + y)) + ((a * math.log(t)) - t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 120.0)
		tmp = Float64(log(z) + Float64(log(y) + Float64(log(t) * Float64(a - 0.5))));
	else
		tmp = Float64(log(Float64(x + y)) + Float64(Float64(a * log(t)) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 120.0)
		tmp = log(z) + (log(y) + (log(t) * (a - 0.5)));
	else
		tmp = log((x + y)) + ((a * log(t)) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 120.0], N[(N[Log[z], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 120

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      2. +-commutative 99.3%

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

      3. associate-+l+ 99.3%

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

      4. +-commutative 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 64.8%

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

   5. Taylor expanded in t around 0 64.0%

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

   if 120 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r- 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around inf 97.4%

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

      1. *-commutative 97.4%

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

   6. Simplified 97.4%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 120:\\ \;\;\;\;\log z + \left(\log y + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \left(a \cdot \log t - t\right)\\ \end{array} \]

Alternative 6: 68.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-+l+ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-def 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 69.0%

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

5. Final simplification 69.0%

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

Alternative 7: 67.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right) - t\\ t_2 := a \cdot \log t\\ \mathbf{if}\;a \leq -6.5 \cdot 10^{-17}:\\ \;\;\;\;\left(\log y + t_2\right) - t\\ \mathbf{elif}\;a \leq -1.1 \cdot 10^{-174}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 8 \cdot 10^{-280}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (log (* y (* z (pow t -0.5)))) t)) (t_2 (* a (log t))))
   (if (<= a -6.5e-17)
     (- (+ (log y) t_2) t)
     (if (<= a -1.1e-174)
       t_1
       (if (<= a 8e-280)
         (- (log (+ x y)) t)
         (if (<= a 8.2e-34) t_1 (+ (- (log z) t) t_2)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((y * (z * pow(t, -0.5)))) - t;
	double t_2 = a * log(t);
	double tmp;
	if (a <= -6.5e-17) {
		tmp = (log(y) + t_2) - t;
	} else if (a <= -1.1e-174) {
		tmp = t_1;
	} else if (a <= 8e-280) {
		tmp = log((x + y)) - t;
	} else if (a <= 8.2e-34) {
		tmp = t_1;
	} else {
		tmp = (log(z) - t) + t_2;
	}
	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((y * (z * (t ** (-0.5d0))))) - t
    t_2 = a * log(t)
    if (a <= (-6.5d-17)) then
        tmp = (log(y) + t_2) - t
    else if (a <= (-1.1d-174)) then
        tmp = t_1
    else if (a <= 8d-280) then
        tmp = log((x + y)) - t
    else if (a <= 8.2d-34) then
        tmp = t_1
    else
        tmp = (log(z) - t) + t_2
    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((y * (z * Math.pow(t, -0.5)))) - t;
	double t_2 = a * Math.log(t);
	double tmp;
	if (a <= -6.5e-17) {
		tmp = (Math.log(y) + t_2) - t;
	} else if (a <= -1.1e-174) {
		tmp = t_1;
	} else if (a <= 8e-280) {
		tmp = Math.log((x + y)) - t;
	} else if (a <= 8.2e-34) {
		tmp = t_1;
	} else {
		tmp = (Math.log(z) - t) + t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log((y * (z * math.pow(t, -0.5)))) - t
	t_2 = a * math.log(t)
	tmp = 0
	if a <= -6.5e-17:
		tmp = (math.log(y) + t_2) - t
	elif a <= -1.1e-174:
		tmp = t_1
	elif a <= 8e-280:
		tmp = math.log((x + y)) - t
	elif a <= 8.2e-34:
		tmp = t_1
	else:
		tmp = (math.log(z) - t) + t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(y * Float64(z * (t ^ -0.5)))) - t)
	t_2 = Float64(a * log(t))
	tmp = 0.0
	if (a <= -6.5e-17)
		tmp = Float64(Float64(log(y) + t_2) - t);
	elseif (a <= -1.1e-174)
		tmp = t_1;
	elseif (a <= 8e-280)
		tmp = Float64(log(Float64(x + y)) - t);
	elseif (a <= 8.2e-34)
		tmp = t_1;
	else
		tmp = Float64(Float64(log(z) - t) + t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log((y * (z * (t ^ -0.5)))) - t;
	t_2 = a * log(t);
	tmp = 0.0;
	if (a <= -6.5e-17)
		tmp = (log(y) + t_2) - t;
	elseif (a <= -1.1e-174)
		tmp = t_1;
	elseif (a <= 8e-280)
		tmp = log((x + y)) - t;
	elseif (a <= 8.2e-34)
		tmp = t_1;
	else
		tmp = (log(z) - t) + t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(y * N[(z * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6.5e-17], N[(N[(N[Log[y], $MachinePrecision] + t$95$2), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, -1.1e-174], t$95$1, If[LessEqual[a, 8e-280], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 8.2e-34], t$95$1, N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -6.4999999999999996e-17

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r- 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 95.6%

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

      1. *-commutative 95.6%

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

   6. Simplified 95.6%

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

   7. Taylor expanded in x around 0 72.3%

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

   if -6.4999999999999996e-17 < a < -1.10000000000000011e-174 or 7.9999999999999997e-280 < a < 8.2000000000000007e-34

   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+ 99.5%

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

      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+ 99.4%

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

      4. +-commutative 99.4%

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

      5. fma-def 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 60.6%

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

   5. Taylor expanded in a around 0 60.6%

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

   6. Taylor expanded in y around inf 60.6%

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

      1. associate-+r+ 60.7%

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

      2. mul-1-neg 60.7%

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

      3. log-rec 60.7%

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

      4. remove-double-neg 60.7%

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

      5. log-prod 42.7%

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

      6. *-commutative 42.7%

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

      7. log-prod 60.7%

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

      8. *-commutative 60.7%

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

      9. associate-+r+ 60.6%

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

      10. rem-log-exp 42.1%

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

      11. exp-sum 39.6%

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

      12. rem-exp-log 39.8%

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

      13. exp-sum 39.8%

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

      14. rem-exp-log 39.9%

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

      15. exp-to-pow 40.0%

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

   8. Simplified 40.0%

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

   if -1.10000000000000011e-174 < a < 7.9999999999999997e-280

   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+ 99.7%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.8%

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

      5. fma-def 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in a around inf 64.6%

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

      1. *-commutative 64.6%

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

   6. Simplified 64.6%

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

   7. Taylor expanded in a around 0 64.6%

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

      1. +-commutative 64.6%

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

   9. Simplified 64.6%

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

   if 8.2000000000000007e-34 < a

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 81.8%

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

   5. Taylor expanded in a around inf 94.1%

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

      1. *-commutative 94.1%

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

   7. Simplified 94.1%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{-17}:\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \mathbf{elif}\;a \leq -1.1 \cdot 10^{-174}:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{elif}\;a \leq 8 \cdot 10^{-280}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{-34}:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 8: 75.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{+15}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(\left(a + -0.5\right) \cdot \log t - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.9e+15)
   (+ (log (* (+ x y) z)) (- (* (+ a -0.5) (log t)) t))
   (- (+ (log y) (* a (log t))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.9e+15) {
		tmp = log(((x + y) * z)) + (((a + -0.5) * log(t)) - t);
	} else {
		tmp = (log(y) + (a * log(t))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 1.9d+15) then
        tmp = log(((x + y) * z)) + (((a + (-0.5d0)) * log(t)) - t)
    else
        tmp = (log(y) + (a * log(t))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.9e+15) {
		tmp = Math.log(((x + y) * z)) + (((a + -0.5) * Math.log(t)) - t);
	} else {
		tmp = (Math.log(y) + (a * Math.log(t))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.9e+15:
		tmp = math.log(((x + y) * z)) + (((a + -0.5) * math.log(t)) - t)
	else:
		tmp = (math.log(y) + (a * math.log(t))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.9e+15)
		tmp = Float64(log(Float64(Float64(x + y) * z)) + Float64(Float64(Float64(a + -0.5) * log(t)) - t));
	else
		tmp = Float64(Float64(log(y) + Float64(a * log(t))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 1.9e+15)
		tmp = log(((x + y) * z)) + (((a + -0.5) * log(t)) - t);
	else
		tmp = (log(y) + (a * log(t))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.9e+15], N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.9e15

   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. cancel-sign-sub 99.4%

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

      2. cancel-sign-sub-inv 99.4%

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

      3. associate--l+ 99.4%

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

      4. remove-double-neg 99.4%

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

      5. sub-neg 99.4%

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

      6. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. associate-+r- 99.4%

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

      2. metadata-eval 99.4%

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

      3. sub-neg 99.4%

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

      4. associate-+l- 99.4%

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

      5. +-commutative 99.4%

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

      6. sum-log 70.5%

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

      7. sub-neg 70.5%

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

      8. metadata-eval 70.5%

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

   5. Applied egg-rr 70.5%

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

   if 1.9e15 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r- 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around inf 99.7%

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

      1. *-commutative 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around 0 73.2%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{+15}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(\left(a + -0.5\right) \cdot \log t - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \end{array} \]

Alternative 9: 60.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{-17} \lor \neg \left(a \leq 3.2 \cdot 10^{-79}\right):\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot z\right) + \left(-0.5 \cdot \log t - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -8.5e-17) (not (<= a 3.2e-79)))
   (- (+ (log y) (* a (log t))) t)
   (+ (log (* y z)) (- (* -0.5 (log t)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -8.5e-17) || !(a <= 3.2e-79)) {
		tmp = (log(y) + (a * log(t))) - t;
	} else {
		tmp = log((y * z)) + ((-0.5 * log(t)) - t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-8.5d-17)) .or. (.not. (a <= 3.2d-79))) then
        tmp = (log(y) + (a * log(t))) - t
    else
        tmp = log((y * z)) + (((-0.5d0) * log(t)) - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -8.5e-17) || !(a <= 3.2e-79)) {
		tmp = (Math.log(y) + (a * Math.log(t))) - t;
	} else {
		tmp = Math.log((y * z)) + ((-0.5 * Math.log(t)) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -8.5e-17) or not (a <= 3.2e-79):
		tmp = (math.log(y) + (a * math.log(t))) - t
	else:
		tmp = math.log((y * z)) + ((-0.5 * math.log(t)) - t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -8.5e-17) || !(a <= 3.2e-79))
		tmp = Float64(Float64(log(y) + Float64(a * log(t))) - t);
	else
		tmp = Float64(log(Float64(y * z)) + Float64(Float64(-0.5 * log(t)) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -8.5e-17) || ~((a <= 3.2e-79)))
		tmp = (log(y) + (a * log(t))) - t;
	else
		tmp = log((y * z)) + ((-0.5 * log(t)) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -8.5e-17], N[Not[LessEqual[a, 3.2e-79]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.5e-17 or 3.19999999999999988e-79 < a

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+r- 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 93.2%

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

      1. *-commutative 93.2%

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

   6. Simplified 93.2%

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

   7. Taylor expanded in x around 0 73.7%

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

   if -8.5e-17 < a < 3.19999999999999988e-79

   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-def 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. Taylor expanded in x around 0 59.9%

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

   5. Taylor expanded in a around 0 59.9%

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

   6. Taylor expanded in t around inf 59.9%

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

      1. associate-+r+ 59.9%

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

      2. log-prod 43.1%

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

      3. *-commutative 43.1%

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

      4. neg-mul-1 43.1%

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

      5. +-commutative 43.1%

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

      6. unsub-neg 43.1%

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

      7. log-rec 43.1%

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

   8. Simplified 43.1%

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

   9. Taylor expanded in t around 0 43.1%

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

       1. neg-mul-1 43.1%

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

       2. +-commutative 43.1%

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

       3. sub-neg 43.1%

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

       4. *-commutative 43.1%

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

   11. Simplified 43.1%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{-17} \lor \neg \left(a \leq 3.2 \cdot 10^{-79}\right):\\ \;\;\;\;\left(\log y + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot z\right) + \left(-0.5 \cdot \log t - t\right)\\ \end{array} \]

Alternative 10: 64.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log z + a \cdot \log t\\ \mathbf{if}\;t \leq 9.5 \cdot 10^{-99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-83}:\\ \;\;\;\;\log \left(y \cdot z\right) + -0.5 \cdot \log t\\ \mathbf{elif}\;t \leq 6200000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log z) (* a (log t)))))
   (if (<= t 9.5e-99)
     t_1
     (if (<= t 2e-83)
       (+ (log (* y z)) (* -0.5 (log t)))
       (if (<= t 6200000000.0) t_1 (- (log (+ x y)) t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(z) + (a * log(t));
	double tmp;
	if (t <= 9.5e-99) {
		tmp = t_1;
	} else if (t <= 2e-83) {
		tmp = log((y * z)) + (-0.5 * log(t));
	} else if (t <= 6200000000.0) {
		tmp = t_1;
	} else {
		tmp = log((x + y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(z) + (a * log(t))
    if (t <= 9.5d-99) then
        tmp = t_1
    else if (t <= 2d-83) then
        tmp = log((y * z)) + ((-0.5d0) * log(t))
    else if (t <= 6200000000.0d0) then
        tmp = t_1
    else
        tmp = log((x + y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log(z) + (a * Math.log(t));
	double tmp;
	if (t <= 9.5e-99) {
		tmp = t_1;
	} else if (t <= 2e-83) {
		tmp = Math.log((y * z)) + (-0.5 * Math.log(t));
	} else if (t <= 6200000000.0) {
		tmp = t_1;
	} else {
		tmp = Math.log((x + y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(z) + (a * math.log(t))
	tmp = 0
	if t <= 9.5e-99:
		tmp = t_1
	elif t <= 2e-83:
		tmp = math.log((y * z)) + (-0.5 * math.log(t))
	elif t <= 6200000000.0:
		tmp = t_1
	else:
		tmp = math.log((x + y)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(z) + Float64(a * log(t)))
	tmp = 0.0
	if (t <= 9.5e-99)
		tmp = t_1;
	elseif (t <= 2e-83)
		tmp = Float64(log(Float64(y * z)) + Float64(-0.5 * log(t)));
	elseif (t <= 6200000000.0)
		tmp = t_1;
	else
		tmp = Float64(log(Float64(x + y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(z) + (a * log(t));
	tmp = 0.0;
	if (t <= 9.5e-99)
		tmp = t_1;
	elseif (t <= 2e-83)
		tmp = log((y * z)) + (-0.5 * log(t));
	elseif (t <= 6200000000.0)
		tmp = t_1;
	else
		tmp = log((x + y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[z], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 9.5e-99], t$95$1, If[LessEqual[t, 2e-83], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6200000000.0], t$95$1, N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 9.5000000000000008e-99 or 2.0000000000000001e-83 < t < 6.2e9

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      2. +-commutative 99.3%

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

      3. associate-+l+ 99.3%

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

      4. +-commutative 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 65.7%

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

   5. Taylor expanded in a around inf 54.9%

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

      1. *-commutative 54.9%

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

   7. Simplified 54.9%

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

   8. Taylor expanded in t around 0 54.6%

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

      1. *-commutative 54.6%

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

   10. Simplified 54.6%

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

   if 9.5000000000000008e-99 < t < 2.0000000000000001e-83

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. fma-def 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 57.4%

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

   5. Taylor expanded in a around 0 44.8%

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

   6. Taylor expanded in t around inf 44.8%

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

      1. associate-+r+ 44.8%

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

      2. log-prod 44.9%

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

      3. *-commutative 44.9%

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

      4. neg-mul-1 44.9%

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

      5. +-commutative 44.9%

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

      6. unsub-neg 44.9%

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

      7. log-rec 44.9%

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

   8. Simplified 44.9%

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

   9. Taylor expanded in t around 0 44.9%

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

       1. +-commutative 44.9%

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

       2. *-commutative 44.9%

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

   11. Simplified 44.9%

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

   if 6.2e9 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r- 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around inf 98.6%

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

      1. *-commutative 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in a around 0 76.6%

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

      1. +-commutative 76.6%

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

   9. Simplified 76.6%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.5 \cdot 10^{-99}:\\ \;\;\;\;\log z + a \cdot \log t\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-83}:\\ \;\;\;\;\log \left(y \cdot z\right) + -0.5 \cdot \log t\\ \mathbf{elif}\;t \leq 6200000000:\\ \;\;\;\;\log z + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \]

Alternative 11: 73.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.00027:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \left(a \cdot \log t - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 0.00027)
   (+ (* (+ a -0.5) (log t)) (log (* y z)))
   (+ (log (+ x y)) (- (* a (log t)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.00027) {
		tmp = ((a + -0.5) * log(t)) + log((y * z));
	} else {
		tmp = log((x + y)) + ((a * log(t)) - t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 0.00027d0) then
        tmp = ((a + (-0.5d0)) * log(t)) + log((y * z))
    else
        tmp = log((x + y)) + ((a * log(t)) - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.00027) {
		tmp = ((a + -0.5) * Math.log(t)) + Math.log((y * z));
	} else {
		tmp = Math.log((x + y)) + ((a * Math.log(t)) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 0.00027:
		tmp = ((a + -0.5) * math.log(t)) + math.log((y * z))
	else:
		tmp = math.log((x + y)) + ((a * math.log(t)) - t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 0.00027)
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) + log(Float64(y * z)));
	else
		tmp = Float64(log(Float64(x + y)) + Float64(Float64(a * log(t)) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 0.00027)
		tmp = ((a + -0.5) * log(t)) + log((y * z));
	else
		tmp = log((x + y)) + ((a * log(t)) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 0.00027], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.70000000000000003e-4

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      2. +-commutative 99.3%

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

      3. associate-+l+ 99.3%

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

      4. +-commutative 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 64.5%

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

   5. Taylor expanded in t around 0 64.1%

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

      1. associate-+r+ 64.1%

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

      2. log-prod 44.7%

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

      3. +-commutative 44.7%

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

      4. sub-neg 44.7%

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

      5. metadata-eval 44.7%

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

   7. Simplified 44.7%

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

   if 2.70000000000000003e-4 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r- 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around inf 96.7%

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

      1. *-commutative 96.7%

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

   6. Simplified 96.7%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.00027:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \left(a \cdot \log t - t\right)\\ \end{array} \]

Alternative 12: 65.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2500000000:\\ \;\;\;\;\log z + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2500000000.0) (+ (log z) (* a (log t))) (- (log (+ x y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2500000000.0) {
		tmp = log(z) + (a * log(t));
	} else {
		tmp = log((x + y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 2500000000.0d0) then
        tmp = log(z) + (a * log(t))
    else
        tmp = log((x + y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2500000000.0) {
		tmp = Math.log(z) + (a * Math.log(t));
	} else {
		tmp = Math.log((x + y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 2500000000.0:
		tmp = math.log(z) + (a * math.log(t))
	else:
		tmp = math.log((x + y)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2500000000.0)
		tmp = Float64(log(z) + Float64(a * log(t)));
	else
		tmp = Float64(log(Float64(x + y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 2500000000.0)
		tmp = log(z) + (a * log(t));
	else
		tmp = log((x + y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2500000000.0], N[(N[Log[z], $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.5e9

   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+ 99.3%

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

      2. +-commutative 99.3%

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

      3. associate-+l+ 99.4%

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

      4. +-commutative 99.4%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 65.3%

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

   5. Taylor expanded in a around inf 52.9%

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

      1. *-commutative 52.9%

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

   7. Simplified 52.9%

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

   8. Taylor expanded in t around 0 52.6%

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

      1. *-commutative 52.6%

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

   10. Simplified 52.6%

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

   if 2.5e9 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r- 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around inf 98.6%

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

      1. *-commutative 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in a around 0 76.6%

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

      1. +-commutative 76.6%

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

   9. Simplified 76.6%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2500000000:\\ \;\;\;\;\log z + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \]

Alternative 13: 77.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. +-commutative 99.6%

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

   3. associate-+l+ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-def 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 69.0%

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

5. Taylor expanded in a around inf 73.1%

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

   1. *-commutative 73.1%

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

7. Simplified 73.1%

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

8. Final simplification 73.1%

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

Alternative 14: 57.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. associate-+l+ 99.6%

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

   3. +-commutative 99.6%

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

   4. associate-+r- 99.6%

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

   5. fma-def 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a around inf 73.8%

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

   1. *-commutative 73.8%

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

6. Simplified 73.8%

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

7. Taylor expanded in x around 0 55.7%

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

8. Final simplification 55.7%

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

Alternative 15: 62.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3400000000:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 3400000000.0) (* a (log t)) (- (log (+ x y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 3400000000.0) {
		tmp = a * log(t);
	} else {
		tmp = log((x + y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 3400000000.0d0) then
        tmp = a * log(t)
    else
        tmp = log((x + y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 3400000000.0) {
		tmp = a * Math.log(t);
	} else {
		tmp = Math.log((x + y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 3400000000.0:
		tmp = a * math.log(t)
	else:
		tmp = math.log((x + y)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 3400000000.0)
		tmp = Float64(a * log(t));
	else
		tmp = Float64(log(Float64(x + y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 3400000000.0)
		tmp = a * log(t);
	else
		tmp = log((x + y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 3400000000.0], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.4e9

   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+ 99.3%

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

      2. associate-+l+ 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r- 99.4%

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

      5. fma-def 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in a around inf 54.3%

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

      1. *-commutative 54.3%

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

   6. Simplified 54.3%

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

   7. Taylor expanded in a around inf 47.2%

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

   if 3.4e9 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r- 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around inf 98.6%

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

      1. *-commutative 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in a around 0 76.6%

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

      1. +-commutative 76.6%

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

   9. Simplified 76.6%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3400000000:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \]

Alternative 16: 62.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2300000000:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2300000000.0) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2300000000.0) {
		tmp = a * log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 2300000000.0d0) then
        tmp = a * log(t)
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2300000000.0) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 2300000000.0:
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2300000000.0)
		tmp = Float64(a * log(t));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 2300000000.0)
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2300000000.0], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 2.3e9

   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+ 99.3%

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

      2. associate-+l+ 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r- 99.4%

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

      5. fma-def 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in a around inf 54.3%

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

      1. *-commutative 54.3%

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

   6. Simplified 54.3%

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

   7. Taylor expanded in a around inf 47.2%

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

   if 2.3e9 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r- 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 76.5%

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

      1. neg-mul-1 76.5%

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

   6. Simplified 76.5%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2300000000:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 17: 62.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 6300000000:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log z - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 6300000000.0) (* a (log t)) (- (log z) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 6300000000.0) {
		tmp = a * log(t);
	} else {
		tmp = 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 <= 6300000000.0d0) then
        tmp = a * log(t)
    else
        tmp = 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 <= 6300000000.0) {
		tmp = a * Math.log(t);
	} else {
		tmp = Math.log(z) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 6300000000.0:
		tmp = a * math.log(t)
	else:
		tmp = math.log(z) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 6300000000.0)
		tmp = Float64(a * log(t));
	else
		tmp = Float64(log(z) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 6300000000.0)
		tmp = a * log(t);
	else
		tmp = log(z) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 6300000000.0], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.3e9

   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+ 99.3%

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

      2. associate-+l+ 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r- 99.4%

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

      5. fma-def 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in a around inf 54.3%

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

      1. *-commutative 54.3%

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

   6. Simplified 54.3%

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

   7. Taylor expanded in a around inf 47.2%

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

   if 6.3e9 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 73.8%

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

   5. Taylor expanded in a around inf 98.5%

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

      1. *-commutative 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in a around 0 76.6%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6300000000:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log z - t\\ \end{array} \]

Alternative 18: 40.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 560:\\ \;\;\;\;\log z\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (if (<= t 560.0) (log z) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 560.0) {
		tmp = log(z);
	} 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 <= 560.0d0) then
        tmp = log(z)
    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 <= 560.0) {
		tmp = Math.log(z);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 560.0:
		tmp = math.log(z)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 560.0)
		tmp = log(z);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 560.0)
		tmp = log(z);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 560.0], N[Log[z], $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 560

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-+l+ 99.4%

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

      4. +-commutative 99.4%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 65.0%

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

   5. Taylor expanded in a around inf 52.9%

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

      1. *-commutative 52.9%

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

   7. Simplified 52.9%

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

   8. Taylor expanded in a around 0 9.2%

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

   9. Taylor expanded in t around 0 9.3%

      \[\leadsto \color{blue}{\log z} \]

   if 560 < t

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r- 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 75.1%

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

      1. neg-mul-1 75.1%

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

   6. Simplified 75.1%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 560:\\ \;\;\;\;\log z\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 19: 37.8% accurate, 156.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. associate-+l+ 99.6%

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

   3. +-commutative 99.6%

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

   4. associate-+r- 99.6%

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

   5. fma-def 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in t around inf 35.5%

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

   1. neg-mul-1 35.5%

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

6. Simplified 35.5%

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

7. Final simplification 35.5%

   \[\leadsto -t \]

Developer target: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023277 
(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))))