Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 19.4s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. 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. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 0.8× speedup

Math

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

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)) + fma(Float64(a + -0.5), log(t), Float64(log(z) - t)))
end

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   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. fma-def 99.6%

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

   5. remove-double-neg 99.6%

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

   6. remove-double-neg 99.6%

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

   7. sub-neg 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 3: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log z \leq 165:\\ \;\;\;\;\left(\left(a + -0.5\right) \cdot \log t + \log \left(z \cdot \left(x + y\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (log z) 165.0)
   (- (+ (* (+ a -0.5) (log t)) (log (* z (+ x y)))) t)
   (- (* a (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (log(z) <= 165.0) {
		tmp = (((a + -0.5) * log(t)) + log((z * (x + y)))) - t;
	} else {
		tmp = (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 (log(z) <= 165.0d0) then
        tmp = (((a + (-0.5d0)) * log(t)) + log((z * (x + y)))) - t
    else
        tmp = (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 (Math.log(z) <= 165.0) {
		tmp = (((a + -0.5) * Math.log(t)) + Math.log((z * (x + y)))) - t;
	} else {
		tmp = (a * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if math.log(z) <= 165.0:
		tmp = (((a + -0.5) * math.log(t)) + math.log((z * (x + y)))) - t
	else:
		tmp = (a * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (log(z) <= 165.0)
		tmp = Float64(Float64(Float64(Float64(a + -0.5) * log(t)) + log(Float64(z * Float64(x + y)))) - t);
	else
		tmp = Float64(Float64(a * log(t)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (log(z) <= 165.0)
		tmp = (((a + -0.5) * log(t)) + log((z * (x + y)))) - t;
	else
		tmp = (a * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[Log[z], $MachinePrecision], 165.0], N[(N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 z) < 165

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. fma-udef 99.5%

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

      3. metadata-eval 99.5%

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

      4. sub-neg 99.5%

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

      5. associate-+r+ 99.5%

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

      6. associate--l+ 99.5%

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

      7. associate-+r- 99.5%

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

      8. sub-neg 99.5%

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

      9. metadata-eval 99.5%

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

      10. +-commutative 99.5%

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

      11. sum-log 95.2%

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

   5. Applied egg-rr 95.2%

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

   if 165 < (log.f64 z)

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. 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. fma-def 99.7%

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

      5. remove-double-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 99.7%

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

   5. Taylor expanded in a around inf 74.5%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z \leq 165:\\ \;\;\;\;\left(\left(a + -0.5\right) \cdot \log t + \log \left(z \cdot \left(x + y\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 4: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 380:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 380.0)
   (+ (log (+ x y)) (+ (log z) (* (log t) (- a 0.5))))
   (- (* a (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 380.0) {
		tmp = log((x + y)) + (log(z) + (log(t) * (a - 0.5)));
	} else {
		tmp = (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 <= 380.0d0) then
        tmp = log((x + y)) + (log(z) + (log(t) * (a - 0.5d0)))
    else
        tmp = (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 <= 380.0) {
		tmp = Math.log((x + y)) + (Math.log(z) + (Math.log(t) * (a - 0.5)));
	} else {
		tmp = (a * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 380.0:
		tmp = math.log((x + y)) + (math.log(z) + (math.log(t) * (a - 0.5)))
	else:
		tmp = (a * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 380.0)
		tmp = Float64(log(Float64(x + y)) + Float64(log(z) + Float64(log(t) * Float64(a - 0.5))));
	else
		tmp = 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 <= 380.0)
		tmp = log((x + y)) + (log(z) + (log(t) * (a - 0.5)));
	else
		tmp = (a * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 380.0], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 380

   1. Initial program 99.2%

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

      1. associate--l+ 99.2%

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

      2. associate-+l+ 99.2%

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

      3. +-commutative 99.2%

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

      4. fma-def 99.2%

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

      5. remove-double-neg 99.2%

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

      6. remove-double-neg 99.2%

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

      7. sub-neg 99.2%

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

      8. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in t around 0 97.9%

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

   if 380 < 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. fma-def 99.9%

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

      5. remove-double-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around inf 99.9%

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

   5. Taylor expanded in a around inf 99.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 380:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 5: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(N[Log[z], $MachinePrecision] + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $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. Final simplification 99.6%

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

Alternative 6: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 350:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \left(\log z + \log y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 350.0)
   (+ (* (log t) (- a 0.5)) (+ (log z) (log y)))
   (- (* a (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 350.0) {
		tmp = (log(t) * (a - 0.5)) + (log(z) + log(y));
	} else {
		tmp = (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 <= 350.0d0) then
        tmp = (log(t) * (a - 0.5d0)) + (log(z) + log(y))
    else
        tmp = (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 <= 350.0) {
		tmp = (Math.log(t) * (a - 0.5)) + (Math.log(z) + Math.log(y));
	} else {
		tmp = (a * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 350.0:
		tmp = (math.log(t) * (a - 0.5)) + (math.log(z) + math.log(y))
	else:
		tmp = (a * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 350.0)
		tmp = Float64(Float64(log(t) * Float64(a - 0.5)) + Float64(log(z) + log(y)));
	else
		tmp = 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 <= 350.0)
		tmp = (log(t) * (a - 0.5)) + (log(z) + log(y));
	else
		tmp = (a * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 350.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 350

   1. Initial program 99.2%

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

      1. associate--l+ 99.2%

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

      2. associate-+l+ 99.2%

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

      3. +-commutative 99.2%

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

      4. fma-def 99.2%

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

      5. remove-double-neg 99.2%

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

      6. remove-double-neg 99.2%

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

      7. sub-neg 99.2%

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

      8. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in t around 0 97.9%

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

   5. Taylor expanded in x around 0 64.1%

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

   if 350 < 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. fma-def 99.9%

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

      5. remove-double-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around inf 99.9%

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

   5. Taylor expanded in a around inf 99.7%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 350:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \left(\log z + \log y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 7: 69.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
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 66.3%

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

5. Final simplification 66.3%

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

Alternative 8: 86.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -200 \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(z \cdot \left(x + y\right)\right) + -0.5 \cdot \log t\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -200.0) (not (<= (- a 0.5) -0.4)))
   (- (* a (log t)) t)
   (- (+ (log (* z (+ x y))) (* -0.5 (log t))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -200.0) || !((a - 0.5) <= -0.4)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = (log((z * (x + y))) + (-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 - 0.5d0) <= (-200.0d0)) .or. (.not. ((a - 0.5d0) <= (-0.4d0)))) then
        tmp = (a * log(t)) - t
    else
        tmp = (log((z * (x + y))) + ((-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 - 0.5) <= -200.0) || !((a - 0.5) <= -0.4)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = (Math.log((z * (x + y))) + (-0.5 * Math.log(t))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((a - 0.5) <= -200.0) or not ((a - 0.5) <= -0.4):
		tmp = (a * math.log(t)) - t
	else:
		tmp = (math.log((z * (x + y))) + (-0.5 * math.log(t))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -200.0) || !(Float64(a - 0.5) <= -0.4))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(Float64(log(Float64(z * Float64(x + y))) + 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 - 0.5) <= -200.0) || ~(((a - 0.5) <= -0.4)))
		tmp = (a * log(t)) - t;
	else
		tmp = (log((z * (x + y))) + (-0.5 * log(t))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -200.0], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a 1/2) < -200 or -0.40000000000000002 < (-.f64 a 1/2)

   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. fma-def 99.7%

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

      5. remove-double-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 99.7%

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

   5. Taylor expanded in a around inf 98.3%

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

   if -200 < (-.f64 a 1/2) < -0.40000000000000002

   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. 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. fma-def 99.4%

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

      5. remove-double-neg 99.4%

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

      6. remove-double-neg 99.4%

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

      7. sub-neg 99.4%

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

      8. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in a around 0 97.9%

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

      1. associate-+r+ 97.9%

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

      2. +-commutative 97.9%

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

      3. associate-+r+ 97.9%

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

      4. +-commutative 97.9%

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

      5. associate-+r+ 97.9%

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

      6. log-prod 76.1%

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

      7. +-commutative 76.1%

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

   6. Simplified 76.1%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -200 \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(z \cdot \left(x + y\right)\right) + -0.5 \cdot \log t\right) - t\\ \end{array} \]

Alternative 9: 73.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -200 \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot \log t + \log \left(z \cdot y\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -200.0) (not (<= (- a 0.5) -0.4)))
   (- (* a (log t)) t)
   (- (+ (* -0.5 (log t)) (log (* z y))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -200.0) || !((a - 0.5) <= -0.4)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = ((-0.5 * log(t)) + log((z * y))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (((a - 0.5d0) <= (-200.0d0)) .or. (.not. ((a - 0.5d0) <= (-0.4d0)))) then
        tmp = (a * log(t)) - t
    else
        tmp = (((-0.5d0) * log(t)) + log((z * y))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -200.0) || !((a - 0.5) <= -0.4)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = ((-0.5 * Math.log(t)) + Math.log((z * y))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((a - 0.5) <= -200.0) or not ((a - 0.5) <= -0.4):
		tmp = (a * math.log(t)) - t
	else:
		tmp = ((-0.5 * math.log(t)) + math.log((z * y))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -200.0) || !(Float64(a - 0.5) <= -0.4))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(Float64(Float64(-0.5 * log(t)) + log(Float64(z * y))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((a - 0.5) <= -200.0) || ~(((a - 0.5) <= -0.4)))
		tmp = (a * log(t)) - t;
	else
		tmp = ((-0.5 * log(t)) + log((z * y))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -200.0], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a 1/2) < -200 or -0.40000000000000002 < (-.f64 a 1/2)

   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. fma-def 99.7%

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

      5. remove-double-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 99.7%

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

   5. Taylor expanded in a around inf 98.3%

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

   if -200 < (-.f64 a 1/2) < -0.40000000000000002

   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. 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. fma-def 99.4%

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

      5. remove-double-neg 99.4%

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

      6. remove-double-neg 99.4%

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

      7. sub-neg 99.4%

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

      8. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in a around inf 99.4%

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

   5. Taylor expanded in a around 0 97.9%

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

      1. associate-+r+ 97.9%

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

      2. metadata-eval 97.9%

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

      3. associate-*r* 97.9%

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

      4. mul-1-neg 97.9%

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

      5. log-rec 97.9%

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

      6. +-commutative 97.9%

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

      7. +-commutative 97.9%

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

      8. log-rec 97.9%

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

      9. mul-1-neg 97.9%

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

      10. associate-*r* 97.9%

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

      11. metadata-eval 97.9%

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

      12. *-commutative 97.9%

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

      13. log-prod 76.1%

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

      14. *-commutative 76.1%

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

      15. +-commutative 76.1%

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

   7. Simplified 76.1%

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

   8. Taylor expanded in x around 0 47.1%

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

      1. *-commutative 47.1%

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

   10. Simplified 47.1%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -200 \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot \log t + \log \left(z \cdot y\right)\right) - t\\ \end{array} \]

Alternative 10: 73.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-24}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \log \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.7e-24)
   (+ (* (+ a -0.5) (log t)) (log (* z y)))
   (- (* a (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.7e-24) {
		tmp = ((a + -0.5) * log(t)) + log((z * y));
	} else {
		tmp = (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 <= 1.7d-24) then
        tmp = ((a + (-0.5d0)) * log(t)) + log((z * y))
    else
        tmp = (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 <= 1.7e-24) {
		tmp = ((a + -0.5) * Math.log(t)) + Math.log((z * y));
	} else {
		tmp = (a * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.7e-24:
		tmp = ((a + -0.5) * math.log(t)) + math.log((z * y))
	else:
		tmp = (a * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.7e-24)
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) + log(Float64(z * y)));
	else
		tmp = 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 <= 1.7e-24)
		tmp = ((a + -0.5) * log(t)) + log((z * y));
	else
		tmp = (a * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.7e-24], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.69999999999999996e-24

   1. Initial program 99.2%

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

      1. associate--l+ 99.2%

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

      2. associate-+l+ 99.2%

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

      3. +-commutative 99.2%

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

      4. fma-def 99.2%

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

      5. remove-double-neg 99.2%

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

      6. remove-double-neg 99.2%

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

      7. sub-neg 99.2%

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

      8. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in t around 0 99.2%

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

   5. Taylor expanded in x around 0 66.2%

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

   6. Taylor expanded in t around inf 66.2%

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

      1. log-prod 49.0%

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

      2. +-commutative 49.0%

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

      3. mul-1-neg 49.0%

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

      4. unsub-neg 49.0%

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

      5. log-rec 49.0%

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

      6. distribute-lft-neg-in 49.0%

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

      7. distribute-rgt-neg-in 49.0%

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

      8. sub-neg 49.0%

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

      9. metadata-eval 49.0%

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

   8. Simplified 49.0%

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

   if 1.69999999999999996e-24 < 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. fma-def 99.9%

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

      5. remove-double-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around inf 99.9%

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

   5. Taylor expanded in a around inf 98.4%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-24}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \log \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 11: 62.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.25 \cdot 10^{+50}:\\ \;\;\;\;t + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 3.25e+50) (+ t (* a (log t))) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 3.25e+50) {
		tmp = t + (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 <= 3.25d+50) then
        tmp = t + (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 <= 3.25e+50) {
		tmp = t + (a * Math.log(t));
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 3.25e+50:
		tmp = t + (a * math.log(t))
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 3.25e+50)
		tmp = Float64(t + 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 <= 3.25e+50)
		tmp = t + (a * log(t));
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 3.25e+50], N[(t + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 3.2500000000000001e50

   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. associate-+l+ 99.3%

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

      3. +-commutative 99.3%

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

      4. fma-def 99.3%

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

      5. remove-double-neg 99.3%

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

      6. remove-double-neg 99.3%

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

      7. sub-neg 99.3%

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

      8. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in a around inf 99.3%

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

   5. Taylor expanded in a around inf 56.1%

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

      1. *-commutative 56.1%

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

      2. fma-neg 56.1%

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

   7. Applied egg-rr 56.1%

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

      1. fma-udef 56.1%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 52.0%

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

      4. sqr-neg 52.0%

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

      5. sqrt-unprod 52.1%

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

      6. add-sqr-sqrt 52.1%

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

   9. Applied egg-rr 52.1%

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

   if 3.2500000000000001e50 < 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. fma-def 99.9%

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

      5. remove-double-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 7.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(x + y\right) + \mathsf{fma}\left(a + -0.5, \log t, \log z - t\right)\right)\right)} \]

      2. +-commutative 7.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(a + -0.5, \log t, \log z - t\right) + \log \left(x + y\right)}\right)\right) \]

   5. Applied egg-rr 7.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(a + -0.5, \log t, \log z - t\right) + \log \left(x + y\right)\right)\right)} \]

   6. Taylor expanded in t around inf 74.9%

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

      1. mul-1-neg 74.9%

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

   8. Simplified 74.9%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.25 \cdot 10^{+50}:\\ \;\;\;\;t + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 12: 62.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{+57}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.8e+57) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.8e+57) {
		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 <= 1.8d+57) 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 <= 1.8e+57) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.8e+57:
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.8e+57)
		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 <= 1.8e+57)
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.8e+57], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 1.8000000000000001e57

   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. associate-+l+ 99.3%

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

      3. +-commutative 99.3%

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

      4. fma-def 99.3%

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

      5. remove-double-neg 99.3%

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

      6. remove-double-neg 99.3%

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

      7. sub-neg 99.3%

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

      8. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in a around inf 99.3%

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

   5. Taylor expanded in a around inf 57.3%

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

   6. Taylor expanded in a around inf 51.9%

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

   if 1.8000000000000001e57 < 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. fma-def 99.9%

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

      5. remove-double-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 7.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(x + y\right) + \mathsf{fma}\left(a + -0.5, \log t, \log z - t\right)\right)\right)} \]

      2. +-commutative 7.1%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(a + -0.5, \log t, \log z - t\right) + \log \left(x + y\right)}\right)\right) \]

   5. Applied egg-rr 7.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(a + -0.5, \log t, \log z - t\right) + \log \left(x + y\right)\right)\right)} \]

   6. Taylor expanded in t around inf 75.8%

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

      1. mul-1-neg 75.8%

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

   8. Simplified 75.8%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{+57}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 13: 74.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ a \cdot \log t - t \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(a * N[Log[t], $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. fma-def 99.6%

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

   5. remove-double-neg 99.6%

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

   6. remove-double-neg 99.6%

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

   7. sub-neg 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a around inf 99.6%

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

5. Taylor expanded in a around inf 76.1%

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

6. Final simplification 76.1%

   \[\leadsto a \cdot \log t - t \]

Alternative 14: 37.6% accurate, 156.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   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. fma-def 99.6%

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

   5. remove-double-neg 99.6%

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

   6. remove-double-neg 99.6%

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

   7. sub-neg 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. expm1-log1p-u 33.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(x + y\right) + \mathsf{fma}\left(a + -0.5, \log t, \log z - t\right)\right)\right)} \]

   2. +-commutative 33.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(a + -0.5, \log t, \log z - t\right) + \log \left(x + y\right)}\right)\right) \]

5. Applied egg-rr 33.0%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(a + -0.5, \log t, \log z - t\right) + \log \left(x + y\right)\right)\right)} \]

6. Taylor expanded in t around inf 37.7%

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

   1. mul-1-neg 37.7%

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

8. Simplified 37.7%

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

9. Final simplification 37.7%

   \[\leadsto -t \]

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 2023196 
(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))))