Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 17.3s

Alternatives: 17

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. associate--l+ 99.6%

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

   3. associate-+r+ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-define 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

Alternative 2: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log z \leq 373:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \left(\log \left(z \cdot \left(x + y\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \log \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (log z) 373.0)
   (+ (* (+ a -0.5) (log t)) (- (log (* z (+ x y))) t))
   (+ (- (log z) t) (log (+ x y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (log(z) <= 373.0) {
		tmp = ((a + -0.5) * log(t)) + (log((z * (x + y))) - t);
	} else {
		tmp = (log(z) - t) + log((x + y));
	}
	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) <= 373.0d0) then
        tmp = ((a + (-0.5d0)) * log(t)) + (log((z * (x + y))) - t)
    else
        tmp = (log(z) - t) + log((x + y))
    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) <= 373.0) {
		tmp = ((a + -0.5) * Math.log(t)) + (Math.log((z * (x + y))) - t);
	} else {
		tmp = (Math.log(z) - t) + Math.log((x + y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if math.log(z) <= 373.0:
		tmp = ((a + -0.5) * math.log(t)) + (math.log((z * (x + y))) - t)
	else:
		tmp = (math.log(z) - t) + math.log((x + y))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (log(z) <= 373.0)
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) + Float64(log(Float64(z * Float64(x + y))) - t));
	else
		tmp = Float64(Float64(log(z) - t) + log(Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (log(z) <= 373.0)
		tmp = ((a + -0.5) * log(t)) + (log((z * (x + y))) - t);
	else
		tmp = (log(z) - t) + log((x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[Log[z], $MachinePrecision], 373.0], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[(N[Log[N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 z) < 373

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. associate-+r- 99.6%

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

      2. sum-log 86.3%

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

   6. Applied egg-rr 86.3%

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

   if 373 < (log.f64 z)

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. associate--l+ 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-undefine 99.8%

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

      8. sub-neg 99.8%

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

      9. +-commutative 99.8%

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

      10. distribute-neg-in 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around inf 50.1%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z \leq 373:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \left(\log \left(z \cdot \left(x + y\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \log \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log z \leq 373:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \left(\log \left(z \cdot y\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \log \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (log z) 373.0)
   (+ (* (+ a -0.5) (log t)) (- (log (* z y)) t))
   (+ (- (log z) t) (log (+ x y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (log(z) <= 373.0) {
		tmp = ((a + -0.5) * log(t)) + (log((z * y)) - t);
	} else {
		tmp = (log(z) - t) + log((x + y));
	}
	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) <= 373.0d0) then
        tmp = ((a + (-0.5d0)) * log(t)) + (log((z * y)) - t)
    else
        tmp = (log(z) - t) + log((x + y))
    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) <= 373.0) {
		tmp = ((a + -0.5) * Math.log(t)) + (Math.log((z * y)) - t);
	} else {
		tmp = (Math.log(z) - t) + Math.log((x + y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if math.log(z) <= 373.0:
		tmp = ((a + -0.5) * math.log(t)) + (math.log((z * y)) - t)
	else:
		tmp = (math.log(z) - t) + math.log((x + y))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (log(z) <= 373.0)
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) + Float64(log(Float64(z * y)) - t));
	else
		tmp = Float64(Float64(log(z) - t) + log(Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (log(z) <= 373.0)
		tmp = ((a + -0.5) * log(t)) + (log((z * y)) - t);
	else
		tmp = (log(z) - t) + log((x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[Log[z], $MachinePrecision], 373.0], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 z) < 373

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 69.6%

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

      1. remove-double-neg 69.6%

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

      2. log-rec 69.6%

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

      3. mul-1-neg 69.6%

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

      4. +-commutative 69.6%

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

      5. associate--l+ 69.6%

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

      6. mul-1-neg 69.6%

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

      7. log-rec 69.6%

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

      8. remove-double-neg 69.6%

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

   7. Simplified 69.6%

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

      1. associate-+r- 69.6%

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

      2. sum-log 59.9%

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

   9. Applied egg-rr 59.9%

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

   if 373 < (log.f64 z)

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. associate--l+ 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-undefine 99.8%

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

      8. sub-neg 99.8%

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

      9. +-commutative 99.8%

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

      10. distribute-neg-in 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around inf 50.1%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z \leq 373:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \left(\log \left(z \cdot y\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \log \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + -0.5\right) \cdot \log t\\ \mathbf{if}\;t \leq 7800000:\\ \;\;\;\;\log z + \left(t\_1 + \log y\right)\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+90}:\\ \;\;\;\;t\_1 + \left(\log \left(z \cdot y\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (+ a -0.5) (log t))))
   (if (<= t 7800000.0)
     (+ (log z) (+ t_1 (log y)))
     (if (<= t 1.5e+90) (+ t_1 (- (log (* z y)) t)) (- t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a + -0.5) * log(t);
	double tmp;
	if (t <= 7800000.0) {
		tmp = log(z) + (t_1 + log(y));
	} else if (t <= 1.5e+90) {
		tmp = t_1 + (log((z * y)) - 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) :: t_1
    real(8) :: tmp
    t_1 = (a + (-0.5d0)) * log(t)
    if (t <= 7800000.0d0) then
        tmp = log(z) + (t_1 + log(y))
    else if (t <= 1.5d+90) then
        tmp = t_1 + (log((z * y)) - 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 t_1 = (a + -0.5) * Math.log(t);
	double tmp;
	if (t <= 7800000.0) {
		tmp = Math.log(z) + (t_1 + Math.log(y));
	} else if (t <= 1.5e+90) {
		tmp = t_1 + (Math.log((z * y)) - t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (a + -0.5) * math.log(t)
	tmp = 0
	if t <= 7800000.0:
		tmp = math.log(z) + (t_1 + math.log(y))
	elif t <= 1.5e+90:
		tmp = t_1 + (math.log((z * y)) - t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a + -0.5) * log(t))
	tmp = 0.0
	if (t <= 7800000.0)
		tmp = Float64(log(z) + Float64(t_1 + log(y)));
	elseif (t <= 1.5e+90)
		tmp = Float64(t_1 + Float64(log(Float64(z * y)) - t));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (a + -0.5) * log(t);
	tmp = 0.0;
	if (t <= 7800000.0)
		tmp = log(z) + (t_1 + log(y));
	elseif (t <= 1.5e+90)
		tmp = t_1 + (log((z * y)) - t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 7800000.0], N[(N[Log[z], $MachinePrecision] + N[(t$95$1 + N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.5e+90], N[(t$95$1 + N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], (-t)]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.8e6

   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+ 99.4%

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

      2. sub-neg 99.4%

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

      3. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 59.9%

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

      1. remove-double-neg 59.9%

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

      2. log-rec 59.9%

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

      3. mul-1-neg 59.9%

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

      4. +-commutative 59.9%

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

      5. associate--l+ 59.9%

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

      6. mul-1-neg 59.9%

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

      7. log-rec 59.9%

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

      8. remove-double-neg 59.9%

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

   7. Simplified 59.9%

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

   8. Taylor expanded in t around 0 58.4%

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

      1. associate-+r+ 58.4%

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

      2. +-commutative 58.4%

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

      3. associate-+r+ 58.4%

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

      4. sub-neg 58.4%

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

      5. metadata-eval 58.4%

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

      6. +-commutative 58.4%

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

   10. Simplified 58.4%

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

   if 7.8e6 < t < 1.49999999999999989e90

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 82.1%

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

      1. remove-double-neg 82.1%

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

      2. log-rec 82.1%

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

      3. mul-1-neg 82.1%

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

      4. +-commutative 82.1%

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

      5. associate--l+ 82.1%

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

      6. mul-1-neg 82.1%

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

      7. log-rec 82.1%

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

      8. remove-double-neg 82.1%

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

   7. Simplified 82.1%

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

      1. associate-+r- 82.1%

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

      2. sum-log 72.6%

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

   9. Applied egg-rr 72.6%

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

   if 1.49999999999999989e90 < t

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. associate--l+ 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-undefine 100.0%

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

      8. sub-neg 100.0%

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

      9. +-commutative 100.0%

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

      10. distribute-neg-in 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 86.3%

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

      1. neg-mul-1 86.3%

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

   7. Simplified 86.3%

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

   8. Taylor expanded in t around inf 86.3%

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

      1. neg-mul-1 86.3%

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

   10. Simplified 86.3%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7800000:\\ \;\;\;\;\log z + \left(\left(a + -0.5\right) \cdot \log t + \log y\right)\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+90}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \left(\log \left(z \cdot y\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 9600000:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+91}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \left(\log \left(z \cdot y\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 9600000.0)
   (+ (log y) (+ (log z) (* (log t) (- a 0.5))))
   (if (<= t 2.7e+91) (+ (* (+ a -0.5) (log t)) (- (log (* z y)) t)) (- t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 9600000.0) {
		tmp = log(y) + (log(z) + (log(t) * (a - 0.5)));
	} else if (t <= 2.7e+91) {
		tmp = ((a + -0.5) * log(t)) + (log((z * y)) - 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 <= 9600000.0d0) then
        tmp = log(y) + (log(z) + (log(t) * (a - 0.5d0)))
    else if (t <= 2.7d+91) then
        tmp = ((a + (-0.5d0)) * log(t)) + (log((z * y)) - 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 <= 9600000.0) {
		tmp = Math.log(y) + (Math.log(z) + (Math.log(t) * (a - 0.5)));
	} else if (t <= 2.7e+91) {
		tmp = ((a + -0.5) * Math.log(t)) + (Math.log((z * y)) - t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 9600000.0:
		tmp = math.log(y) + (math.log(z) + (math.log(t) * (a - 0.5)))
	elif t <= 2.7e+91:
		tmp = ((a + -0.5) * math.log(t)) + (math.log((z * y)) - t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 9600000.0)
		tmp = Float64(log(y) + Float64(log(z) + Float64(log(t) * Float64(a - 0.5))));
	elseif (t <= 2.7e+91)
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) + Float64(log(Float64(z * y)) - t));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 9600000.0)
		tmp = log(y) + (log(z) + (log(t) * (a - 0.5)));
	elseif (t <= 2.7e+91)
		tmp = ((a + -0.5) * log(t)) + (log((z * y)) - t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 9600000.0], N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.7e+91], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], (-t)]]

Derivation

1. Split input into 3 regimes

2. if t < 9.6e6

   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+ 99.4%

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

      2. sub-neg 99.4%

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

      3. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 59.9%

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

      1. remove-double-neg 59.9%

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

      2. log-rec 59.9%

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

      3. mul-1-neg 59.9%

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

      4. +-commutative 59.9%

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

      5. associate--l+ 59.9%

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

      6. mul-1-neg 59.9%

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

      7. log-rec 59.9%

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

      8. remove-double-neg 59.9%

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

   7. Simplified 59.9%

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

   8. Taylor expanded in t around 0 58.4%

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

   if 9.6e6 < t < 2.7e91

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 82.1%

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

      1. remove-double-neg 82.1%

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

      2. log-rec 82.1%

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

      3. mul-1-neg 82.1%

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

      4. +-commutative 82.1%

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

      5. associate--l+ 82.1%

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

      6. mul-1-neg 82.1%

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

      7. log-rec 82.1%

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

      8. remove-double-neg 82.1%

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

   7. Simplified 82.1%

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

      1. associate-+r- 82.1%

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

      2. sum-log 72.6%

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

   9. Applied egg-rr 72.6%

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

   if 2.7e91 < t

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. associate--l+ 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-undefine 100.0%

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

      8. sub-neg 100.0%

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

      9. +-commutative 100.0%

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

      10. distribute-neg-in 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 86.3%

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

      1. neg-mul-1 86.3%

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

   7. Simplified 86.3%

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

   8. Taylor expanded in t around inf 86.3%

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

      1. neg-mul-1 86.3%

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

   10. Simplified 86.3%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9600000:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+91}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \left(\log \left(z \cdot y\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. sub-neg 99.6%

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

   3. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 7: 68.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return ((a + -0.5) * log(t)) + (log(z) + (log(y) - 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 + (-0.5d0)) * log(t)) + (log(z) + (log(y) - t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   3. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 68.2%

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

   1. remove-double-neg 68.2%

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

   2. log-rec 68.2%

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

   3. mul-1-neg 68.2%

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

   4. +-commutative 68.2%

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

   5. associate--l+ 68.2%

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

   6. mul-1-neg 68.2%

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

   7. log-rec 68.2%

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

   8. remove-double-neg 68.2%

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

7. Simplified 68.2%

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

8. Final simplification 68.2%

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

Alternative 8: 60.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right)\\ \mathbf{if}\;a \leq -2.5 \cdot 10^{+21}:\\ \;\;\;\;\log \left(\sqrt[3]{t}\right) \cdot \left(a \cdot 3\right)\\ \mathbf{elif}\;a \leq -6.8 \cdot 10^{-56}:\\ \;\;\;\;\left|t\_1\right| - t\\ \mathbf{elif}\;a \leq 19:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{\left(a + -0.5\right)}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1 + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ x y))))
   (if (<= a -2.5e+21)
     (* (log (cbrt t)) (* a 3.0))
     (if (<= a -6.8e-56)
       (- (fabs t_1) t)
       (if (<= a 19.0)
         (- (log (* y (* z (pow t (+ a -0.5))))) t)
         (+ t_1 (* a (log t))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y));
	double tmp;
	if (a <= -2.5e+21) {
		tmp = log(cbrt(t)) * (a * 3.0);
	} else if (a <= -6.8e-56) {
		tmp = fabs(t_1) - t;
	} else if (a <= 19.0) {
		tmp = log((y * (z * pow(t, (a + -0.5))))) - t;
	} else {
		tmp = t_1 + (a * log(t));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log((x + y));
	double tmp;
	if (a <= -2.5e+21) {
		tmp = Math.log(Math.cbrt(t)) * (a * 3.0);
	} else if (a <= -6.8e-56) {
		tmp = Math.abs(t_1) - t;
	} else if (a <= 19.0) {
		tmp = Math.log((y * (z * Math.pow(t, (a + -0.5))))) - t;
	} else {
		tmp = t_1 + (a * Math.log(t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	tmp = 0.0
	if (a <= -2.5e+21)
		tmp = Float64(log(cbrt(t)) * Float64(a * 3.0));
	elseif (a <= -6.8e-56)
		tmp = Float64(abs(t_1) - t);
	elseif (a <= 19.0)
		tmp = Float64(log(Float64(y * Float64(z * (t ^ Float64(a + -0.5))))) - t);
	else
		tmp = Float64(t_1 + Float64(a * log(t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a, -2.5e+21], N[(N[Log[N[Power[t, 1/3], $MachinePrecision]], $MachinePrecision] * N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -6.8e-56], N[(N[Abs[t$95$1], $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 19.0], N[(N[Log[N[(y * N[(z * N[Power[t, N[(a + -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], N[(t$95$1 + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -2.5e21

   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. sub-neg 99.7%

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

      3. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 99.7%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)} \]

      2. log-prod 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]

      3. pow2 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \left(\log \color{blue}{\left({\left(\sqrt[3]{t}\right)}^{2}\right)} + \log \left(\sqrt[3]{t}\right)\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\left(\log \left({\left(\sqrt[3]{t}\right)}^{2}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]
   7. Step-by-step derivation

      1. log-pow 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{t}\right)} + \log \left(\sqrt[3]{t}\right)\right) \]

      2. fma-define 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\mathsf{fma}\left(2, \log \left(\sqrt[3]{t}\right), \log \left(\sqrt[3]{t}\right)\right)} \]

   8. Applied egg-rr 99.6%

      \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\mathsf{fma}\left(2, \log \left(\sqrt[3]{t}\right), \log \left(\sqrt[3]{t}\right)\right)} \]
   9. Step-by-step derivation

      1. fma-define 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]

      2. distribute-lft1-in 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{t}\right)\right)} \]

      3. metadata-eval 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \left(\color{blue}{3} \cdot \log \left(\sqrt[3]{t}\right)\right) \]

   10. Simplified 99.6%

       \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\left(3 \cdot \log \left(\sqrt[3]{t}\right)\right)} \]

   11. Taylor expanded in a around 0 99.5%

       \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\left(-1.5 \cdot \log \left(\sqrt[3]{t}\right) + 3 \cdot \left(a \cdot \log \left(\sqrt[3]{t}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r* 99.7%

          \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(-1.5 \cdot \log \left(\sqrt[3]{t}\right) + \color{blue}{\left(3 \cdot a\right) \cdot \log \left(\sqrt[3]{t}\right)}\right) \]

       2. distribute-rgt-out 99.7%

          \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\log \left(\sqrt[3]{t}\right) \cdot \left(-1.5 + 3 \cdot a\right)} \]

   13. Simplified 99.7%

       \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\log \left(\sqrt[3]{t}\right) \cdot \left(-1.5 + 3 \cdot a\right)} \]

   14. Taylor expanded in a around inf 76.3%

       \[\leadsto \color{blue}{3 \cdot \left(a \cdot \log \left(\sqrt[3]{t}\right)\right)} \]
   15. Step-by-step derivation

       1. associate-*r* 76.5%

          \[\leadsto \color{blue}{\left(3 \cdot a\right) \cdot \log \left(\sqrt[3]{t}\right)} \]

       2. *-commutative 76.5%

          \[\leadsto \color{blue}{\log \left(\sqrt[3]{t}\right) \cdot \left(3 \cdot a\right)} \]

   16. Simplified 76.5%

       \[\leadsto \color{blue}{\log \left(\sqrt[3]{t}\right) \cdot \left(3 \cdot a\right)} \]

   if -2.5e21 < a < -6.79999999999999964e-56

   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- 99.8%

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

      2. associate--l+ 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-undefine 99.8%

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

      8. sub-neg 99.8%

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

      9. +-commutative 99.8%

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

      10. distribute-neg-in 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around inf 70.6%

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

      1. neg-mul-1 70.6%

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

   7. Simplified 70.6%

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

      1. add-sqr-sqrt 57.9%

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

      2. sqrt-unprod 71.7%

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

      3. pow2 71.7%

         \[\leadsto \sqrt{\color{blue}{{\log \left(x + y\right)}^{2}}} + \left(-t\right) \]

   9. Applied egg-rr 71.7%

      \[\leadsto \color{blue}{\sqrt{{\log \left(x + y\right)}^{2}}} + \left(-t\right) \]
   10. Step-by-step derivation

       1. unpow2 71.7%

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

       2. rem-sqrt-square 71.7%

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

       3. +-commutative 71.7%

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

   11. Simplified 71.7%

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

   if -6.79999999999999964e-56 < a < 19

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. sub-neg 99.5%

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

      3. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 66.6%

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

      1. remove-double-neg 66.6%

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

      2. log-rec 66.6%

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

      3. mul-1-neg 66.6%

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

      4. +-commutative 66.6%

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

      5. associate--l+ 66.6%

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

      6. mul-1-neg 66.6%

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

      7. log-rec 66.6%

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

      8. remove-double-neg 66.6%

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

   7. Simplified 66.6%

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

      1. add-log-exp 14.6%

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

      2. exp-sum 13.9%

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

      3. exp-sum 13.9%

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

      4. add-exp-log 14.0%

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

      5. exp-diff 14.0%

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

      6. add-exp-log 14.3%

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

      7. *-commutative 14.3%

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

      8. exp-to-pow 14.3%

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

   9. Applied egg-rr 14.3%

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

   10. Taylor expanded in t around 0 48.4%

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

       1. *-commutative 48.4%

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

       2. exp-to-pow 48.4%

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

       3. sub-neg 48.4%

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

       4. metadata-eval 48.4%

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

       5. *-commutative 48.4%

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

       6. neg-mul-1 48.4%

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

       7. unsub-neg 48.4%

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

   12. Simplified 48.4%

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

   if 19 < a

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-undefine 99.7%

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

      8. sub-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. distribute-neg-in 99.7%

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

      11. metadata-eval 99.7%

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

      12. metadata-eval 99.7%

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

      13. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around inf 85.1%

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

      1. *-commutative 85.1%

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

   7. Simplified 85.1%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.5 \cdot 10^{+21}:\\ \;\;\;\;\log \left(\sqrt[3]{t}\right) \cdot \left(a \cdot 3\right)\\ \mathbf{elif}\;a \leq -6.8 \cdot 10^{-56}:\\ \;\;\;\;\left|\log \left(x + y\right)\right| - t\\ \mathbf{elif}\;a \leq 19:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{\left(a + -0.5\right)}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right)\\ \mathbf{if}\;a \leq -2.3 \cdot 10^{+20}:\\ \;\;\;\;\log \left(\sqrt[3]{t}\right) \cdot \left(a \cdot 3\right)\\ \mathbf{elif}\;a \leq 410000000:\\ \;\;\;\;\left(\log z - t\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ x y))))
   (if (<= a -2.3e+20)
     (* (log (cbrt t)) (* a 3.0))
     (if (<= a 410000000.0) (+ (- (log z) t) t_1) (+ t_1 (* a (log t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y));
	double tmp;
	if (a <= -2.3e+20) {
		tmp = log(cbrt(t)) * (a * 3.0);
	} else if (a <= 410000000.0) {
		tmp = (log(z) - t) + t_1;
	} else {
		tmp = t_1 + (a * log(t));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log((x + y));
	double tmp;
	if (a <= -2.3e+20) {
		tmp = Math.log(Math.cbrt(t)) * (a * 3.0);
	} else if (a <= 410000000.0) {
		tmp = (Math.log(z) - t) + t_1;
	} else {
		tmp = t_1 + (a * Math.log(t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	tmp = 0.0
	if (a <= -2.3e+20)
		tmp = Float64(log(cbrt(t)) * Float64(a * 3.0));
	elseif (a <= 410000000.0)
		tmp = Float64(Float64(log(z) - t) + t_1);
	else
		tmp = Float64(t_1 + Float64(a * log(t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a, -2.3e+20], N[(N[Log[N[Power[t, 1/3], $MachinePrecision]], $MachinePrecision] * N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 410000000.0], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision], N[(t$95$1 + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.3e20

   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. sub-neg 99.7%

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

      3. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 99.7%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)} \]

      2. log-prod 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]

      3. pow2 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \left(\log \color{blue}{\left({\left(\sqrt[3]{t}\right)}^{2}\right)} + \log \left(\sqrt[3]{t}\right)\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\left(\log \left({\left(\sqrt[3]{t}\right)}^{2}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]
   7. Step-by-step derivation

      1. log-pow 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{t}\right)} + \log \left(\sqrt[3]{t}\right)\right) \]

      2. fma-define 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\mathsf{fma}\left(2, \log \left(\sqrt[3]{t}\right), \log \left(\sqrt[3]{t}\right)\right)} \]

   8. Applied egg-rr 99.6%

      \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\mathsf{fma}\left(2, \log \left(\sqrt[3]{t}\right), \log \left(\sqrt[3]{t}\right)\right)} \]
   9. Step-by-step derivation

      1. fma-define 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]

      2. distribute-lft1-in 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{t}\right)\right)} \]

      3. metadata-eval 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \left(\color{blue}{3} \cdot \log \left(\sqrt[3]{t}\right)\right) \]

   10. Simplified 99.6%

       \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\left(3 \cdot \log \left(\sqrt[3]{t}\right)\right)} \]

   11. Taylor expanded in a around 0 99.5%

       \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\left(-1.5 \cdot \log \left(\sqrt[3]{t}\right) + 3 \cdot \left(a \cdot \log \left(\sqrt[3]{t}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r* 99.7%

          \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(-1.5 \cdot \log \left(\sqrt[3]{t}\right) + \color{blue}{\left(3 \cdot a\right) \cdot \log \left(\sqrt[3]{t}\right)}\right) \]

       2. distribute-rgt-out 99.7%

          \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\log \left(\sqrt[3]{t}\right) \cdot \left(-1.5 + 3 \cdot a\right)} \]

   13. Simplified 99.7%

       \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\log \left(\sqrt[3]{t}\right) \cdot \left(-1.5 + 3 \cdot a\right)} \]

   14. Taylor expanded in a around inf 76.3%

       \[\leadsto \color{blue}{3 \cdot \left(a \cdot \log \left(\sqrt[3]{t}\right)\right)} \]
   15. Step-by-step derivation

       1. associate-*r* 76.5%

          \[\leadsto \color{blue}{\left(3 \cdot a\right) \cdot \log \left(\sqrt[3]{t}\right)} \]

       2. *-commutative 76.5%

          \[\leadsto \color{blue}{\log \left(\sqrt[3]{t}\right) \cdot \left(3 \cdot a\right)} \]

   16. Simplified 76.5%

       \[\leadsto \color{blue}{\log \left(\sqrt[3]{t}\right) \cdot \left(3 \cdot a\right)} \]

   if -2.3e20 < a < 4.1e8

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-undefine 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in t around inf 62.3%

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

   if 4.1e8 < a

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-undefine 99.7%

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

      8. sub-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. distribute-neg-in 99.7%

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

      11. metadata-eval 99.7%

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

      12. metadata-eval 99.7%

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

      13. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around inf 85.1%

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

      1. *-commutative 85.1%

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

   7. Simplified 85.1%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{+20}:\\ \;\;\;\;\log \left(\sqrt[3]{t}\right) \cdot \left(a \cdot 3\right)\\ \mathbf{elif}\;a \leq 410000000:\\ \;\;\;\;\left(\log z - t\right) + \log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right)\\ \mathbf{if}\;a \leq -5.5 \cdot 10^{+20}:\\ \;\;\;\;\log \left(\sqrt[3]{t}\right) \cdot \left(a \cdot 3\right)\\ \mathbf{elif}\;a \leq 1900000000:\\ \;\;\;\;t\_1 - t\\ \mathbf{else}:\\ \;\;\;\;t\_1 + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (log (+ x y))))
   (if (<= a -5.5e+20)
     (* (log (cbrt t)) (* a 3.0))
     (if (<= a 1900000000.0) (- t_1 t) (+ t_1 (* a (log t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y));
	double tmp;
	if (a <= -5.5e+20) {
		tmp = log(cbrt(t)) * (a * 3.0);
	} else if (a <= 1900000000.0) {
		tmp = t_1 - t;
	} else {
		tmp = t_1 + (a * log(t));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log((x + y));
	double tmp;
	if (a <= -5.5e+20) {
		tmp = Math.log(Math.cbrt(t)) * (a * 3.0);
	} else if (a <= 1900000000.0) {
		tmp = t_1 - t;
	} else {
		tmp = t_1 + (a * Math.log(t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	tmp = 0.0
	if (a <= -5.5e+20)
		tmp = Float64(log(cbrt(t)) * Float64(a * 3.0));
	elseif (a <= 1900000000.0)
		tmp = Float64(t_1 - t);
	else
		tmp = Float64(t_1 + Float64(a * log(t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a, -5.5e+20], N[(N[Log[N[Power[t, 1/3], $MachinePrecision]], $MachinePrecision] * N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1900000000.0], N[(t$95$1 - t), $MachinePrecision], N[(t$95$1 + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.5e20

   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. sub-neg 99.7%

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

      3. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 99.7%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)} \]

      2. log-prod 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]

      3. pow2 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \left(\log \color{blue}{\left({\left(\sqrt[3]{t}\right)}^{2}\right)} + \log \left(\sqrt[3]{t}\right)\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\left(\log \left({\left(\sqrt[3]{t}\right)}^{2}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]
   7. Step-by-step derivation

      1. log-pow 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{t}\right)} + \log \left(\sqrt[3]{t}\right)\right) \]

      2. fma-define 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\mathsf{fma}\left(2, \log \left(\sqrt[3]{t}\right), \log \left(\sqrt[3]{t}\right)\right)} \]

   8. Applied egg-rr 99.6%

      \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\mathsf{fma}\left(2, \log \left(\sqrt[3]{t}\right), \log \left(\sqrt[3]{t}\right)\right)} \]
   9. Step-by-step derivation

      1. fma-define 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]

      2. distribute-lft1-in 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{t}\right)\right)} \]

      3. metadata-eval 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \left(\color{blue}{3} \cdot \log \left(\sqrt[3]{t}\right)\right) \]

   10. Simplified 99.6%

       \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\left(3 \cdot \log \left(\sqrt[3]{t}\right)\right)} \]

   11. Taylor expanded in a around 0 99.5%

       \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\left(-1.5 \cdot \log \left(\sqrt[3]{t}\right) + 3 \cdot \left(a \cdot \log \left(\sqrt[3]{t}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r* 99.7%

          \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(-1.5 \cdot \log \left(\sqrt[3]{t}\right) + \color{blue}{\left(3 \cdot a\right) \cdot \log \left(\sqrt[3]{t}\right)}\right) \]

       2. distribute-rgt-out 99.7%

          \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\log \left(\sqrt[3]{t}\right) \cdot \left(-1.5 + 3 \cdot a\right)} \]

   13. Simplified 99.7%

       \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\log \left(\sqrt[3]{t}\right) \cdot \left(-1.5 + 3 \cdot a\right)} \]

   14. Taylor expanded in a around inf 76.3%

       \[\leadsto \color{blue}{3 \cdot \left(a \cdot \log \left(\sqrt[3]{t}\right)\right)} \]
   15. Step-by-step derivation

       1. associate-*r* 76.5%

          \[\leadsto \color{blue}{\left(3 \cdot a\right) \cdot \log \left(\sqrt[3]{t}\right)} \]

       2. *-commutative 76.5%

          \[\leadsto \color{blue}{\log \left(\sqrt[3]{t}\right) \cdot \left(3 \cdot a\right)} \]

   16. Simplified 76.5%

       \[\leadsto \color{blue}{\log \left(\sqrt[3]{t}\right) \cdot \left(3 \cdot a\right)} \]

   if -5.5e20 < a < 1.9e9

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-undefine 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in t around inf 60.3%

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

      1. neg-mul-1 60.3%

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

   7. Simplified 60.3%

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

   if 1.9e9 < a

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-undefine 99.7%

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

      8. sub-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. distribute-neg-in 99.7%

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

      11. metadata-eval 99.7%

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

      12. metadata-eval 99.7%

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

      13. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around inf 85.1%

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

      1. *-commutative 85.1%

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

   7. Simplified 85.1%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{+20}:\\ \;\;\;\;\log \left(\sqrt[3]{t}\right) \cdot \left(a \cdot 3\right)\\ \mathbf{elif}\;a \leq 1900000000:\\ \;\;\;\;\log \left(x + y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+20}:\\ \;\;\;\;\log \left(\sqrt[3]{t}\right) \cdot \left(a \cdot 3\right)\\ \mathbf{elif}\;a \leq 220000000:\\ \;\;\;\;\log \left(x + y\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -3.8e+20)
   (* (log (cbrt t)) (* a 3.0))
   (if (<= a 220000000.0) (- (log (+ x y)) t) (* a (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -3.8e+20) {
		tmp = log(cbrt(t)) * (a * 3.0);
	} else if (a <= 220000000.0) {
		tmp = log((x + y)) - t;
	} else {
		tmp = a * log(t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -3.8e+20) {
		tmp = Math.log(Math.cbrt(t)) * (a * 3.0);
	} else if (a <= 220000000.0) {
		tmp = Math.log((x + y)) - t;
	} else {
		tmp = a * Math.log(t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -3.8e+20)
		tmp = Float64(log(cbrt(t)) * Float64(a * 3.0));
	elseif (a <= 220000000.0)
		tmp = Float64(log(Float64(x + y)) - t);
	else
		tmp = Float64(a * log(t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -3.8e+20], N[(N[Log[N[Power[t, 1/3], $MachinePrecision]], $MachinePrecision] * N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 220000000.0], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.8e20

   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. sub-neg 99.7%

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

      3. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 99.7%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)} \]

      2. log-prod 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]

      3. pow2 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \left(\log \color{blue}{\left({\left(\sqrt[3]{t}\right)}^{2}\right)} + \log \left(\sqrt[3]{t}\right)\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\left(\log \left({\left(\sqrt[3]{t}\right)}^{2}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]
   7. Step-by-step derivation

      1. log-pow 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{t}\right)} + \log \left(\sqrt[3]{t}\right)\right) \]

      2. fma-define 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\mathsf{fma}\left(2, \log \left(\sqrt[3]{t}\right), \log \left(\sqrt[3]{t}\right)\right)} \]

   8. Applied egg-rr 99.6%

      \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\mathsf{fma}\left(2, \log \left(\sqrt[3]{t}\right), \log \left(\sqrt[3]{t}\right)\right)} \]
   9. Step-by-step derivation

      1. fma-define 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{t}\right) + \log \left(\sqrt[3]{t}\right)\right)} \]

      2. distribute-lft1-in 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{t}\right)\right)} \]

      3. metadata-eval 99.6%

         \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \left(\color{blue}{3} \cdot \log \left(\sqrt[3]{t}\right)\right) \]

   10. Simplified 99.6%

       \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \color{blue}{\left(3 \cdot \log \left(\sqrt[3]{t}\right)\right)} \]

   11. Taylor expanded in a around 0 99.5%

       \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\left(-1.5 \cdot \log \left(\sqrt[3]{t}\right) + 3 \cdot \left(a \cdot \log \left(\sqrt[3]{t}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r* 99.7%

          \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(-1.5 \cdot \log \left(\sqrt[3]{t}\right) + \color{blue}{\left(3 \cdot a\right) \cdot \log \left(\sqrt[3]{t}\right)}\right) \]

       2. distribute-rgt-out 99.7%

          \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\log \left(\sqrt[3]{t}\right) \cdot \left(-1.5 + 3 \cdot a\right)} \]

   13. Simplified 99.7%

       \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \color{blue}{\log \left(\sqrt[3]{t}\right) \cdot \left(-1.5 + 3 \cdot a\right)} \]

   14. Taylor expanded in a around inf 76.3%

       \[\leadsto \color{blue}{3 \cdot \left(a \cdot \log \left(\sqrt[3]{t}\right)\right)} \]
   15. Step-by-step derivation

       1. associate-*r* 76.5%

          \[\leadsto \color{blue}{\left(3 \cdot a\right) \cdot \log \left(\sqrt[3]{t}\right)} \]

       2. *-commutative 76.5%

          \[\leadsto \color{blue}{\log \left(\sqrt[3]{t}\right) \cdot \left(3 \cdot a\right)} \]

   16. Simplified 76.5%

       \[\leadsto \color{blue}{\log \left(\sqrt[3]{t}\right) \cdot \left(3 \cdot a\right)} \]

   if -3.8e20 < a < 2.2e8

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-undefine 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in t around inf 60.3%

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

      1. neg-mul-1 60.3%

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

   7. Simplified 60.3%

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

   if 2.2e8 < a

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 65.5%

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

      1. remove-double-neg 65.5%

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

      2. log-rec 65.5%

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

      3. mul-1-neg 65.5%

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

      4. +-commutative 65.5%

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

      5. associate--l+ 65.5%

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

      6. mul-1-neg 65.5%

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

      7. log-rec 65.5%

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

      8. remove-double-neg 65.5%

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

   7. Simplified 65.5%

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

   8. Taylor expanded in a around inf 84.8%

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

      1. *-commutative 84.8%

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

   10. Simplified 84.8%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+20}:\\ \;\;\;\;\log \left(\sqrt[3]{t}\right) \cdot \left(a \cdot 3\right)\\ \mathbf{elif}\;a \leq 220000000:\\ \;\;\;\;\log \left(x + y\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.2 \cdot 10^{+21} \lor \neg \left(a \leq 2400000000\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -7.2e+21) (not (<= a 2400000000.0)))
   (* a (log t))
   (- (log (+ x y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -7.2e+21) || !(a <= 2400000000.0)) {
		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 ((a <= (-7.2d+21)) .or. (.not. (a <= 2400000000.0d0))) 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 ((a <= -7.2e+21) || !(a <= 2400000000.0)) {
		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 (a <= -7.2e+21) or not (a <= 2400000000.0):
		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 ((a <= -7.2e+21) || !(a <= 2400000000.0))
		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 ((a <= -7.2e+21) || ~((a <= 2400000000.0)))
		tmp = a * log(t);
	else
		tmp = log((x + y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -7.2e+21], N[Not[LessEqual[a, 2400000000.0]], $MachinePrecision]], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.2e21 or 2.4e9 < 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. sub-neg 99.7%

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

      3. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 69.7%

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

      1. remove-double-neg 69.7%

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

      2. log-rec 69.7%

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

      3. mul-1-neg 69.7%

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

      4. +-commutative 69.7%

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

      5. associate--l+ 69.7%

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

      6. mul-1-neg 69.7%

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

      7. log-rec 69.7%

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

      8. remove-double-neg 69.7%

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

   7. Simplified 69.7%

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

   8. Taylor expanded in a around inf 81.0%

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

      1. *-commutative 81.0%

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

   10. Simplified 81.0%

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

   if -7.2e21 < a < 2.4e9

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-undefine 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in t around inf 60.3%

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

      1. neg-mul-1 60.3%

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

   7. Simplified 60.3%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.2 \cdot 10^{+21} \lor \neg \left(a \leq 2400000000\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.8 \cdot 10^{+20} \lor \neg \left(a \leq 540000000\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.8e+20) (not (<= a 540000000.0)))
   (* a (log t))
   (- (log y) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.8e+20) || !(a <= 540000000.0)) {
		tmp = a * log(t);
	} else {
		tmp = log(y) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-1.8d+20)) .or. (.not. (a <= 540000000.0d0))) then
        tmp = a * log(t)
    else
        tmp = log(y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.8e+20) || !(a <= 540000000.0)) {
		tmp = a * Math.log(t);
	} else {
		tmp = Math.log(y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -1.8e+20) or not (a <= 540000000.0):
		tmp = a * math.log(t)
	else:
		tmp = math.log(y) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.8e+20) || !(a <= 540000000.0))
		tmp = Float64(a * log(t));
	else
		tmp = Float64(log(y) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -1.8e+20) || ~((a <= 540000000.0)))
		tmp = a * log(t);
	else
		tmp = log(y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.8e+20], N[Not[LessEqual[a, 540000000.0]], $MachinePrecision]], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.8e20 or 5.4e8 < 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. sub-neg 99.7%

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

      3. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 69.7%

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

      1. remove-double-neg 69.7%

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

      2. log-rec 69.7%

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

      3. mul-1-neg 69.7%

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

      4. +-commutative 69.7%

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

      5. associate--l+ 69.7%

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

      6. mul-1-neg 69.7%

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

      7. log-rec 69.7%

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

      8. remove-double-neg 69.7%

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

   7. Simplified 69.7%

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

   8. Taylor expanded in a around inf 81.0%

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

      1. *-commutative 81.0%

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

   10. Simplified 81.0%

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

   if -1.8e20 < a < 5.4e8

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-undefine 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in t around inf 60.3%

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

      1. neg-mul-1 60.3%

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

   7. Simplified 60.3%

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

   8. Taylor expanded in x around 0 49.7%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.8 \cdot 10^{+20} \lor \neg \left(a \leq 540000000\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{+72}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 4.5e+72) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 4.5e+72) {
		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 <= 4.5d+72) 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 <= 4.5e+72) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 4.5e+72:
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 4.5e+72)
		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 <= 4.5e+72)
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 4.5e+72], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 4.4999999999999998e72

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. sub-neg 99.4%

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

      3. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 62.2%

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

      1. remove-double-neg 62.2%

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

      2. log-rec 62.2%

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

      3. mul-1-neg 62.2%

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

      4. +-commutative 62.2%

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

      5. associate--l+ 62.2%

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

      6. mul-1-neg 62.2%

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

      7. log-rec 62.2%

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

      8. remove-double-neg 62.2%

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

   7. Simplified 62.2%

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

   8. Taylor expanded in a around inf 54.9%

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

      1. *-commutative 54.9%

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

   10. Simplified 54.9%

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

   if 4.4999999999999998e72 < t

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. associate--l+ 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-undefine 100.0%

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

      8. sub-neg 100.0%

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

      9. +-commutative 100.0%

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

      10. distribute-neg-in 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 83.8%

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

      1. neg-mul-1 83.8%

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

   7. Simplified 83.8%

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

   8. Taylor expanded in t around inf 83.8%

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

      1. neg-mul-1 83.8%

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

   10. Simplified 83.8%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{+72}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 41.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 510:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{1}{t} + -1\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 510.0) (log (+ x y)) (+ (* t (+ (/ 1.0 t) -1.0)) -1.0)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 510.0) {
		tmp = log((x + y));
	} else {
		tmp = (t * ((1.0 / t) + -1.0)) + -1.0;
	}
	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 <= 510.0d0) then
        tmp = log((x + y))
    else
        tmp = (t * ((1.0d0 / t) + (-1.0d0))) + (-1.0d0)
    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 <= 510.0) {
		tmp = Math.log((x + y));
	} else {
		tmp = (t * ((1.0 / t) + -1.0)) + -1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 510.0:
		tmp = math.log((x + y))
	else:
		tmp = (t * ((1.0 / t) + -1.0)) + -1.0
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 510.0)
		tmp = log(Float64(x + y));
	else
		tmp = Float64(Float64(t * Float64(Float64(1.0 / t) + -1.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 510.0)
		tmp = log((x + y));
	else
		tmp = (t * ((1.0 / t) + -1.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 510.0], N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision], N[(N[(t * N[(N[(1.0 / t), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 510

   1. Initial program 99.3%

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

      1. associate-+l- 99.3%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. fma-undefine 99.3%

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

      8. sub-neg 99.3%

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

      9. +-commutative 99.3%

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

      10. distribute-neg-in 99.3%

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

      11. metadata-eval 99.3%

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

      12. metadata-eval 99.3%

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

      13. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in t around inf 8.5%

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

      1. neg-mul-1 8.5%

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

   7. Simplified 8.5%

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

   8. Taylor expanded in t around 0 8.5%

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

      1. +-commutative 8.5%

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

   10. Simplified 8.5%

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

   if 510 < t

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. associate--l+ 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-undefine 99.9%

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

      8. sub-neg 99.9%

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

      9. +-commutative 99.9%

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

      10. distribute-neg-in 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 74.8%

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

      1. neg-mul-1 74.8%

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

   7. Simplified 74.8%

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

   8. Taylor expanded in t around inf 74.8%

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

      1. neg-mul-1 74.8%

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

   10. Simplified 74.8%

       \[\leadsto \color{blue}{-t} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 0.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-t\right)\right)} \]

       2. expm1-undefine 0.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]

   12. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]
   13. Step-by-step derivation

       1. sub-neg 0.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} + \left(-1\right)} \]

       2. log1p-undefine 0.0%

          \[\leadsto e^{\color{blue}{\log \left(1 + \left(-t\right)\right)}} + \left(-1\right) \]

       3. rem-exp-log 74.8%

          \[\leadsto \color{blue}{\left(1 + \left(-t\right)\right)} + \left(-1\right) \]

       4. unsub-neg 74.8%

          \[\leadsto \color{blue}{\left(1 - t\right)} + \left(-1\right) \]

       5. metadata-eval 74.8%

          \[\leadsto \left(1 - t\right) + \color{blue}{-1} \]

   14. Simplified 74.8%

       \[\leadsto \color{blue}{\left(1 - t\right) + -1} \]

   15. Taylor expanded in t around inf 74.8%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 510:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{1}{t} + -1\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 37.7% accurate, 34.8× speedup

Math

\[\begin{array}{l} \\ t \cdot \left(\frac{1}{t} + -1\right) + -1 \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ (* t (+ (/ 1.0 t) -1.0)) -1.0))

C

double code(double x, double y, double z, double t, double a) {
	return (t * ((1.0 / t) + -1.0)) + -1.0;
}

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 * ((1.0d0 / t) + (-1.0d0))) + (-1.0d0)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return (t * ((1.0 / t) + -1.0)) + -1.0;
}

Python

def code(x, y, z, t, a):
	return (t * ((1.0 / t) + -1.0)) + -1.0

Julia

function code(x, y, z, t, a)
	return Float64(Float64(t * Float64(Float64(1.0 / t) + -1.0)) + -1.0)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = (t * ((1.0 / t) + -1.0)) + -1.0;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(t * N[(N[(1.0 / t), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. associate-+l- 99.6%

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

   2. associate--l+ 99.6%

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

   3. sub-neg 99.6%

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

   4. +-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. distribute-rgt-neg-in 99.6%

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

   7. fma-undefine 99.6%

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

   8. sub-neg 99.6%

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

   9. +-commutative 99.6%

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

   10. distribute-neg-in 99.6%

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

   11. metadata-eval 99.6%

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

   12. metadata-eval 99.6%

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

   13. unsub-neg 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in t around inf 41.1%

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

   1. neg-mul-1 41.1%

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

7. Simplified 41.1%

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

8. Taylor expanded in t around inf 38.2%

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

   1. neg-mul-1 38.2%

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

10. Simplified 38.2%

    \[\leadsto \color{blue}{-t} \]
11. Step-by-step derivation

    1. expm1-log1p-u 1.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-t\right)\right)} \]

    2. expm1-undefine 1.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]

12. Applied egg-rr 1.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]
13. Step-by-step derivation

    1. sub-neg 1.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} + \left(-1\right)} \]

    2. log1p-undefine 1.4%

       \[\leadsto e^{\color{blue}{\log \left(1 + \left(-t\right)\right)}} + \left(-1\right) \]

    3. rem-exp-log 38.2%

       \[\leadsto \color{blue}{\left(1 + \left(-t\right)\right)} + \left(-1\right) \]

    4. unsub-neg 38.2%

       \[\leadsto \color{blue}{\left(1 - t\right)} + \left(-1\right) \]

    5. metadata-eval 38.2%

       \[\leadsto \left(1 - t\right) + \color{blue}{-1} \]

14. Simplified 38.2%

    \[\leadsto \color{blue}{\left(1 - t\right) + -1} \]

15. Taylor expanded in t around inf 38.3%

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

16. Final simplification 38.3%

    \[\leadsto t \cdot \left(\frac{1}{t} + -1\right) + -1 \]
17. Add Preprocessing

Alternative 17: 37.7% accurate, 156.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-+l- 99.6%

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

   2. associate--l+ 99.6%

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

   3. sub-neg 99.6%

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

   4. +-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. distribute-rgt-neg-in 99.6%

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

   7. fma-undefine 99.6%

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

   8. sub-neg 99.6%

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

   9. +-commutative 99.6%

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

   10. distribute-neg-in 99.6%

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

   11. metadata-eval 99.6%

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

   12. metadata-eval 99.6%

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

   13. unsub-neg 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in t around inf 41.1%

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

   1. neg-mul-1 41.1%

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

7. Simplified 41.1%

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

8. Taylor expanded in t around inf 38.2%

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

   1. neg-mul-1 38.2%

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

10. Simplified 38.2%

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

Developer Target 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024137 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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