Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 24.1s

Alternatives: 17

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

Alternative 2: 76.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log z \leq 517.2:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(\log t \cdot \left(a + -0.5\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (log z) 517.2)
   (+ (log (* (+ x y) z)) (- (* (log t) (+ a -0.5)) t))
   (- (+ (log z) (log (* y (pow t -0.5)))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (log(z) <= 517.2) {
		tmp = log(((x + y) * z)) + ((log(t) * (a + -0.5)) - t);
	} else {
		tmp = (log(z) + log((y * pow(t, -0.5)))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (log(z) <= 517.2d0) then
        tmp = log(((x + y) * z)) + ((log(t) * (a + (-0.5d0))) - t)
    else
        tmp = (log(z) + log((y * (t ** (-0.5d0))))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (Math.log(z) <= 517.2) {
		tmp = Math.log(((x + y) * z)) + ((Math.log(t) * (a + -0.5)) - t);
	} else {
		tmp = (Math.log(z) + Math.log((y * Math.pow(t, -0.5)))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if math.log(z) <= 517.2:
		tmp = math.log(((x + y) * z)) + ((math.log(t) * (a + -0.5)) - t)
	else:
		tmp = (math.log(z) + math.log((y * math.pow(t, -0.5)))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (log(z) <= 517.2)
		tmp = Float64(log(Float64(Float64(x + y) * z)) + Float64(Float64(log(t) * Float64(a + -0.5)) - t));
	else
		tmp = Float64(Float64(log(z) + log(Float64(y * (t ^ -0.5)))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (log(z) <= 517.2)
		tmp = log(((x + y) * z)) + ((log(t) * (a + -0.5)) - t);
	else
		tmp = (log(z) + log((y * (t ^ -0.5)))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[Log[z], $MachinePrecision], 517.2], N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[Log[N[(y * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 z) < 517.20000000000005

   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. remove-double-neg 99.5%

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

      2. associate--l+ 99.5%

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

      3. remove-double-neg 99.5%

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

      4. sub-neg 99.5%

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

      5. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. metadata-eval 99.5%

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

      2. sub-neg 99.5%

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

      3. associate-+r- 99.5%

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

      4. associate-+l- 99.5%

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

      5. +-commutative 99.5%

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

      6. sum-log 86.9%

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

      7. sub-neg 86.9%

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

      8. metadata-eval 86.9%

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

      9. *-commutative 86.9%

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

   6. Applied egg-rr 86.9%

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

      1. +-commutative 86.9%

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

   8. Simplified 86.9%

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

   if 517.20000000000005 < (log.f64 z)

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. associate--l+ 99.8%

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

      3. remove-double-neg 99.8%

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

      4. sub-neg 99.8%

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

      5. 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 a around 0 66.8%

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

   6. Taylor expanded in x around 0 34.6%

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

      1. +-commutative 34.6%

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

      2. *-un-lft-identity 34.6%

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

      3. fma-define 34.6%

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

      4. add-log-exp 34.6%

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

      5. sum-log 30.6%

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

      6. *-commutative 30.6%

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

      7. exp-to-pow 30.6%

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

   8. Applied egg-rr 30.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, \log \left(y \cdot {t}^{-0.5}\right), \log z\right)} - t \]
   9. Step-by-step derivation

      1. fma-undefine 30.6%

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

      2. *-lft-identity 30.6%

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

   10. Simplified 30.6%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z \leq 517.2:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(\log t \cdot \left(a + -0.5\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot a\\ \mathbf{if}\;a - 0.5 \leq -2 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a - 0.5 \leq 10:\\ \;\;\;\;\left(\log y + \left(\log z + \log t \cdot -0.5\right)\right) - t\\ \mathbf{elif}\;a - 0.5 \leq 5 \cdot 10^{+128}:\\ \;\;\;\;\log \left(y \cdot z\right) + \mathsf{fma}\left(\log t, a + -0.5, -t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) a)))
   (if (<= (- a 0.5) -2e+21)
     t_1
     (if (<= (- a 0.5) 10.0)
       (- (+ (log y) (+ (log z) (* (log t) -0.5))) t)
       (if (<= (- a 0.5) 5e+128)
         (+ (log (* y z)) (fma (log t) (+ a -0.5) (- t)))
         t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * a;
	double tmp;
	if ((a - 0.5) <= -2e+21) {
		tmp = t_1;
	} else if ((a - 0.5) <= 10.0) {
		tmp = (log(y) + (log(z) + (log(t) * -0.5))) - t;
	} else if ((a - 0.5) <= 5e+128) {
		tmp = log((y * z)) + fma(log(t), (a + -0.5), -t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * a)
	tmp = 0.0
	if (Float64(a - 0.5) <= -2e+21)
		tmp = t_1;
	elseif (Float64(a - 0.5) <= 10.0)
		tmp = Float64(Float64(log(y) + Float64(log(z) + Float64(log(t) * -0.5))) - t);
	elseif (Float64(a - 0.5) <= 5e+128)
		tmp = Float64(log(Float64(y * z)) + fma(log(t), Float64(a + -0.5), Float64(-t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -2e+21], t$95$1, If[LessEqual[N[(a - 0.5), $MachinePrecision], 10.0], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(a - 0.5), $MachinePrecision], 5e+128], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + (-t)), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -2e21 or 5e128 < (-.f64 a #s(literal 1/2 binary64))

   1. Initial program 99.6%

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

      1. expm1-log1p-u 44.3%

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

      2. expm1-undefine 44.3%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]

   4. Applied egg-rr 44.3%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-define 44.3%

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

   6. Simplified 44.3%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 44.3%

         \[\leadsto \left(\color{blue}{1 \cdot \left(\log \left(x + y\right) + \log z\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      2. +-commutative 44.3%

         \[\leadsto \left(1 \cdot \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      3. sum-log 33.3%

         \[\leadsto \left(1 \cdot \color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

   8. Applied egg-rr 33.3%

      \[\leadsto \left(\color{blue}{1 \cdot \log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]
   9. Step-by-step derivation

      1. *-lft-identity 33.3%

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

      2. +-commutative 33.3%

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

   10. Simplified 33.3%

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

   11. Taylor expanded in a around inf 83.2%

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

       1. *-commutative 83.2%

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

   13. Simplified 83.2%

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

   if -2e21 < (-.f64 a #s(literal 1/2 binary64)) < 10

   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. remove-double-neg 99.5%

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

      2. associate--l+ 99.5%

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

      3. remove-double-neg 99.5%

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

      4. sub-neg 99.5%

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

      5. 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 a around 0 97.9%

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

   6. Taylor expanded in x around 0 59.3%

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

      1. +-commutative 59.3%

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

   8. Simplified 59.3%

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

   if 10 < (-.f64 a #s(literal 1/2 binary64)) < 5e128

   1. Initial program 99.5%

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

      1. expm1-log1p-u 55.3%

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

      2. expm1-undefine 55.3%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]

   4. Applied egg-rr 55.3%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-define 55.3%

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

   6. Simplified 55.3%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 55.3%

         \[\leadsto \left(\color{blue}{1 \cdot \left(\log \left(x + y\right) + \log z\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      2. +-commutative 55.3%

         \[\leadsto \left(1 \cdot \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      3. sum-log 44.2%

         \[\leadsto \left(1 \cdot \color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

   8. Applied egg-rr 44.2%

      \[\leadsto \left(\color{blue}{1 \cdot \log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]
   9. Step-by-step derivation

      1. *-lft-identity 44.2%

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

      2. +-commutative 44.2%

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

   10. Simplified 44.2%

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

   11. Taylor expanded in x around 0 62.8%

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

       1. associate--l+ 62.8%

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

       2. sub-neg 62.8%

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

       3. metadata-eval 62.8%

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

       4. +-commutative 62.8%

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

       5. distribute-rgt-out 62.8%

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

       6. +-commutative 62.8%

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

       7. distribute-rgt-in 62.8%

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

       8. fma-neg 63.0%

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

       9. +-commutative 63.0%

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

   13. Simplified 63.0%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -2 \cdot 10^{+21}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{elif}\;a - 0.5 \leq 10:\\ \;\;\;\;\left(\log y + \left(\log z + \log t \cdot -0.5\right)\right) - t\\ \mathbf{elif}\;a - 0.5 \leq 5 \cdot 10^{+128}:\\ \;\;\;\;\log \left(y \cdot z\right) + \mathsf{fma}\left(\log t, a + -0.5, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot a\\ \mathbf{if}\;a - 0.5 \leq -2 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a - 0.5 \leq 10:\\ \;\;\;\;\left(\log y + \left(\log z + \log t \cdot -0.5\right)\right) - t\\ \mathbf{elif}\;a - 0.5 \leq 5 \cdot 10^{+128}:\\ \;\;\;\;\left(\log t \cdot \left(a - 0.5\right) + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) a)))
   (if (<= (- a 0.5) -2e+21)
     t_1
     (if (<= (- a 0.5) 10.0)
       (- (+ (log y) (+ (log z) (* (log t) -0.5))) t)
       (if (<= (- a 0.5) 5e+128)
         (- (+ (* (log t) (- a 0.5)) (log (* y z))) t)
         t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * a;
	double tmp;
	if ((a - 0.5) <= -2e+21) {
		tmp = t_1;
	} else if ((a - 0.5) <= 10.0) {
		tmp = (log(y) + (log(z) + (log(t) * -0.5))) - t;
	} else if ((a - 0.5) <= 5e+128) {
		tmp = ((log(t) * (a - 0.5)) + log((y * z))) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(t) * a
    if ((a - 0.5d0) <= (-2d+21)) then
        tmp = t_1
    else if ((a - 0.5d0) <= 10.0d0) then
        tmp = (log(y) + (log(z) + (log(t) * (-0.5d0)))) - t
    else if ((a - 0.5d0) <= 5d+128) then
        tmp = ((log(t) * (a - 0.5d0)) + log((y * z))) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log(t) * a;
	double tmp;
	if ((a - 0.5) <= -2e+21) {
		tmp = t_1;
	} else if ((a - 0.5) <= 10.0) {
		tmp = (Math.log(y) + (Math.log(z) + (Math.log(t) * -0.5))) - t;
	} else if ((a - 0.5) <= 5e+128) {
		tmp = ((Math.log(t) * (a - 0.5)) + Math.log((y * z))) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(t) * a
	tmp = 0
	if (a - 0.5) <= -2e+21:
		tmp = t_1
	elif (a - 0.5) <= 10.0:
		tmp = (math.log(y) + (math.log(z) + (math.log(t) * -0.5))) - t
	elif (a - 0.5) <= 5e+128:
		tmp = ((math.log(t) * (a - 0.5)) + math.log((y * z))) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * a)
	tmp = 0.0
	if (Float64(a - 0.5) <= -2e+21)
		tmp = t_1;
	elseif (Float64(a - 0.5) <= 10.0)
		tmp = Float64(Float64(log(y) + Float64(log(z) + Float64(log(t) * -0.5))) - t);
	elseif (Float64(a - 0.5) <= 5e+128)
		tmp = Float64(Float64(Float64(log(t) * Float64(a - 0.5)) + log(Float64(y * z))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(t) * a;
	tmp = 0.0;
	if ((a - 0.5) <= -2e+21)
		tmp = t_1;
	elseif ((a - 0.5) <= 10.0)
		tmp = (log(y) + (log(z) + (log(t) * -0.5))) - t;
	elseif ((a - 0.5) <= 5e+128)
		tmp = ((log(t) * (a - 0.5)) + log((y * z))) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[N[(a - 0.5), $MachinePrecision], -2e+21], t$95$1, If[LessEqual[N[(a - 0.5), $MachinePrecision], 10.0], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(a - 0.5), $MachinePrecision], 5e+128], N[(N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -2e21 or 5e128 < (-.f64 a #s(literal 1/2 binary64))

   1. Initial program 99.6%

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

      1. expm1-log1p-u 44.3%

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

      2. expm1-undefine 44.3%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]

   4. Applied egg-rr 44.3%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-define 44.3%

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

   6. Simplified 44.3%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 44.3%

         \[\leadsto \left(\color{blue}{1 \cdot \left(\log \left(x + y\right) + \log z\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      2. +-commutative 44.3%

         \[\leadsto \left(1 \cdot \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      3. sum-log 33.3%

         \[\leadsto \left(1 \cdot \color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

   8. Applied egg-rr 33.3%

      \[\leadsto \left(\color{blue}{1 \cdot \log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]
   9. Step-by-step derivation

      1. *-lft-identity 33.3%

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

      2. +-commutative 33.3%

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

   10. Simplified 33.3%

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

   11. Taylor expanded in a around inf 83.2%

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

       1. *-commutative 83.2%

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

   13. Simplified 83.2%

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

   if -2e21 < (-.f64 a #s(literal 1/2 binary64)) < 10

   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. remove-double-neg 99.5%

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

      2. associate--l+ 99.5%

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

      3. remove-double-neg 99.5%

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

      4. sub-neg 99.5%

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

      5. 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 a around 0 97.9%

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

   6. Taylor expanded in x around 0 59.3%

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

      1. +-commutative 59.3%

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

   8. Simplified 59.3%

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

   if 10 < (-.f64 a #s(literal 1/2 binary64)) < 5e128

   1. Initial program 99.5%

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

      1. expm1-log1p-u 55.3%

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

      2. expm1-undefine 55.3%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]

   4. Applied egg-rr 55.3%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-define 55.3%

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

   6. Simplified 55.3%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 55.3%

         \[\leadsto \left(\color{blue}{1 \cdot \left(\log \left(x + y\right) + \log z\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      2. +-commutative 55.3%

         \[\leadsto \left(1 \cdot \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      3. sum-log 44.2%

         \[\leadsto \left(1 \cdot \color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

   8. Applied egg-rr 44.2%

      \[\leadsto \left(\color{blue}{1 \cdot \log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]
   9. Step-by-step derivation

      1. *-lft-identity 44.2%

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

      2. +-commutative 44.2%

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

   10. Simplified 44.2%

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

   11. Taylor expanded in x around 0 62.8%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -2 \cdot 10^{+21}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{elif}\;a - 0.5 \leq 10:\\ \;\;\;\;\left(\log y + \left(\log z + \log t \cdot -0.5\right)\right) - t\\ \mathbf{elif}\;a - 0.5 \leq 5 \cdot 10^{+128}:\\ \;\;\;\;\left(\log t \cdot \left(a - 0.5\right) + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. remove-double-neg 99.6%

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

   2. associate--l+ 99.6%

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

   3. remove-double-neg 99.6%

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

   4. sub-neg 99.6%

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

   5. 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%

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

Alternative 6: 68.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. remove-double-neg 99.6%

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

   2. associate--l+ 99.6%

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

   3. remove-double-neg 99.6%

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

   4. sub-neg 99.6%

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

   5. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 67.8%

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

Alternative 7: 59.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right) - t\\ t_2 := \log t \cdot a\\ t_3 := \left(\log z + \log y\right) - t\\ \mathbf{if}\;a \leq -700000:\\ \;\;\;\;\log \left(x + y\right) + t\_2\\ \mathbf{elif}\;a \leq -1.1 \cdot 10^{-267}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.9 \cdot 10^{-271}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.85 \cdot 10^{+50}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (log (* y (* z (pow t -0.5)))) t))
        (t_2 (* (log t) a))
        (t_3 (- (+ (log z) (log y)) t)))
   (if (<= a -700000.0)
     (+ (log (+ x y)) t_2)
     (if (<= a -1.1e-267)
       t_1
       (if (<= a 3.9e-271)
         t_3
         (if (<= a 1.7e-47) t_1 (if (<= a 1.85e+50) t_3 t_2)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((y * (z * pow(t, -0.5)))) - t;
	double t_2 = log(t) * a;
	double t_3 = (log(z) + log(y)) - t;
	double tmp;
	if (a <= -700000.0) {
		tmp = log((x + y)) + t_2;
	} else if (a <= -1.1e-267) {
		tmp = t_1;
	} else if (a <= 3.9e-271) {
		tmp = t_3;
	} else if (a <= 1.7e-47) {
		tmp = t_1;
	} else if (a <= 1.85e+50) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = log((y * (z * (t ** (-0.5d0))))) - t
    t_2 = log(t) * a
    t_3 = (log(z) + log(y)) - t
    if (a <= (-700000.0d0)) then
        tmp = log((x + y)) + t_2
    else if (a <= (-1.1d-267)) then
        tmp = t_1
    else if (a <= 3.9d-271) then
        tmp = t_3
    else if (a <= 1.7d-47) then
        tmp = t_1
    else if (a <= 1.85d+50) then
        tmp = t_3
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log((y * (z * Math.pow(t, -0.5)))) - t;
	double t_2 = Math.log(t) * a;
	double t_3 = (Math.log(z) + Math.log(y)) - t;
	double tmp;
	if (a <= -700000.0) {
		tmp = Math.log((x + y)) + t_2;
	} else if (a <= -1.1e-267) {
		tmp = t_1;
	} else if (a <= 3.9e-271) {
		tmp = t_3;
	} else if (a <= 1.7e-47) {
		tmp = t_1;
	} else if (a <= 1.85e+50) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log((y * (z * math.pow(t, -0.5)))) - t
	t_2 = math.log(t) * a
	t_3 = (math.log(z) + math.log(y)) - t
	tmp = 0
	if a <= -700000.0:
		tmp = math.log((x + y)) + t_2
	elif a <= -1.1e-267:
		tmp = t_1
	elif a <= 3.9e-271:
		tmp = t_3
	elif a <= 1.7e-47:
		tmp = t_1
	elif a <= 1.85e+50:
		tmp = t_3
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(y * Float64(z * (t ^ -0.5)))) - t)
	t_2 = Float64(log(t) * a)
	t_3 = Float64(Float64(log(z) + log(y)) - t)
	tmp = 0.0
	if (a <= -700000.0)
		tmp = Float64(log(Float64(x + y)) + t_2);
	elseif (a <= -1.1e-267)
		tmp = t_1;
	elseif (a <= 3.9e-271)
		tmp = t_3;
	elseif (a <= 1.7e-47)
		tmp = t_1;
	elseif (a <= 1.85e+50)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log((y * (z * (t ^ -0.5)))) - t;
	t_2 = log(t) * a;
	t_3 = (log(z) + log(y)) - t;
	tmp = 0.0;
	if (a <= -700000.0)
		tmp = log((x + y)) + t_2;
	elseif (a <= -1.1e-267)
		tmp = t_1;
	elseif (a <= 3.9e-271)
		tmp = t_3;
	elseif (a <= 1.7e-47)
		tmp = t_1;
	elseif (a <= 1.85e+50)
		tmp = t_3;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(y * N[(z * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[a, -700000.0], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[a, -1.1e-267], t$95$1, If[LessEqual[a, 3.9e-271], t$95$3, If[LessEqual[a, 1.7e-47], t$95$1, If[LessEqual[a, 1.85e+50], t$95$3, t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -7e5

   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 a around inf 79.9%

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

      1. *-commutative 79.9%

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

   7. Simplified 79.9%

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

   if -7e5 < a < -1.09999999999999994e-267 or 3.89999999999999997e-271 < a < 1.7000000000000001e-47

   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. remove-double-neg 99.5%

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

      2. associate--l+ 99.5%

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

      3. remove-double-neg 99.5%

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

      4. sub-neg 99.5%

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

      5. 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 a around 0 99.2%

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

   6. Taylor expanded in x around 0 61.4%

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

      1. +-commutative 61.4%

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

   8. Simplified 61.4%

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

      1. add-log-exp 55.2%

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

      2. sum-log 42.7%

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

      3. +-commutative 42.7%

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

      4. exp-sum 42.6%

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

      5. add-exp-log 42.7%

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

      6. *-commutative 42.7%

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

      7. exp-to-pow 42.8%

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

   10. Applied egg-rr 42.8%

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

   if -1.09999999999999994e-267 < a < 3.89999999999999997e-271 or 1.7000000000000001e-47 < a < 1.85e50

   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%

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

      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 70.2%

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

   6. Taylor expanded in x around 0 43.5%

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

   if 1.85e50 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. expm1-log1p-u 49.5%

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

      2. expm1-undefine 49.4%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]

   4. Applied egg-rr 49.4%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-define 49.5%

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

   6. Simplified 49.5%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 49.5%

         \[\leadsto \left(\color{blue}{1 \cdot \left(\log \left(x + y\right) + \log z\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      2. +-commutative 49.5%

         \[\leadsto \left(1 \cdot \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      3. sum-log 37.6%

         \[\leadsto \left(1 \cdot \color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

   8. Applied egg-rr 37.6%

      \[\leadsto \left(\color{blue}{1 \cdot \log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]
   9. Step-by-step derivation

      1. *-lft-identity 37.6%

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

      2. +-commutative 37.6%

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

   10. Simplified 37.6%

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

   11. Taylor expanded in a around inf 80.8%

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

       1. *-commutative 80.8%

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

   13. Simplified 80.8%

       \[\leadsto \color{blue}{\log t \cdot a} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -700000:\\ \;\;\;\;\log \left(x + y\right) + \log t \cdot a\\ \mathbf{elif}\;a \leq -1.1 \cdot 10^{-267}:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{elif}\;a \leq 3.9 \cdot 10^{-271}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-47}:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{elif}\;a \leq 1.85 \cdot 10^{+50}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{+163}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(\log t \cdot \left(a + -0.5\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 9e+163)
   (+ (log (* (+ x y) z)) (- (* (log t) (+ a -0.5)) t))
   (+ (- 1.0 t) -1.0)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 9e+163) {
		tmp = log(((x + y) * z)) + ((log(t) * (a + -0.5)) - t);
	} else {
		tmp = (1.0 - t) + -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 <= 9d+163) then
        tmp = log(((x + y) * z)) + ((log(t) * (a + (-0.5d0))) - t)
    else
        tmp = (1.0d0 - t) + (-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 <= 9e+163) {
		tmp = Math.log(((x + y) * z)) + ((Math.log(t) * (a + -0.5)) - t);
	} else {
		tmp = (1.0 - t) + -1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 9e+163:
		tmp = math.log(((x + y) * z)) + ((math.log(t) * (a + -0.5)) - t)
	else:
		tmp = (1.0 - t) + -1.0
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 9e+163)
		tmp = Float64(log(Float64(Float64(x + y) * z)) + Float64(Float64(log(t) * Float64(a + -0.5)) - t));
	else
		tmp = Float64(Float64(1.0 - t) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 9e+163)
		tmp = log(((x + y) * z)) + ((log(t) * (a + -0.5)) - t);
	else
		tmp = (1.0 - t) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 9e+163], N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.99999999999999976e163

   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. remove-double-neg 99.4%

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

      2. associate--l+ 99.4%

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

      3. remove-double-neg 99.4%

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

      4. sub-neg 99.4%

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

      5. 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. Step-by-step derivation

      1. metadata-eval 99.4%

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

      2. sub-neg 99.4%

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

      3. associate-+r- 99.4%

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

      4. associate-+l- 99.4%

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

      5. +-commutative 99.4%

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

      6. sum-log 78.9%

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

      7. sub-neg 78.9%

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

      8. metadata-eval 78.9%

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

      9. *-commutative 78.9%

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

   6. Applied egg-rr 78.9%

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

      1. +-commutative 78.9%

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

   8. Simplified 78.9%

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

   if 8.99999999999999976e163 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. expm1-log1p-u 99.8%

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

      2. expm1-undefine 99.8%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-define 99.8%

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

   6. Simplified 99.8%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 99.8%

         \[\leadsto \left(\color{blue}{1 \cdot \left(\log \left(x + y\right) + \log z\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      2. +-commutative 99.8%

         \[\leadsto \left(1 \cdot \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      3. sum-log 67.5%

         \[\leadsto \left(1 \cdot \color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

   8. Applied egg-rr 67.5%

      \[\leadsto \left(\color{blue}{1 \cdot \log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]
   9. Step-by-step derivation

      1. *-lft-identity 67.5%

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

      2. +-commutative 67.5%

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

   10. Simplified 67.5%

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

   11. Taylor expanded in t around inf 87.8%

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

       1. neg-mul-1 87.8%

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

   13. Simplified 87.8%

       \[\leadsto \color{blue}{-t} \]
   14. 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%

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

   15. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]
   16. 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 87.8%

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

       4. unsub-neg 87.8%

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

       5. metadata-eval 87.8%

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

   17. Simplified 87.8%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{+163}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(\log t \cdot \left(a + -0.5\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{+18} \lor \neg \left(a \leq 1.6 \cdot 10^{+50}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -3.1e+18) (not (<= a 1.6e+50)))
   (* (log t) a)
   (+ (log (+ x y)) (- (log z) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3.1e+18) || !(a <= 1.6e+50)) {
		tmp = log(t) * a;
	} else {
		tmp = log((x + y)) + (log(z) - t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-3.1d+18)) .or. (.not. (a <= 1.6d+50))) then
        tmp = log(t) * a
    else
        tmp = log((x + y)) + (log(z) - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3.1e+18) || !(a <= 1.6e+50)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = Math.log((x + y)) + (Math.log(z) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -3.1e+18) or not (a <= 1.6e+50):
		tmp = math.log(t) * a
	else:
		tmp = math.log((x + y)) + (math.log(z) - t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -3.1e+18) || !(a <= 1.6e+50))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(log(Float64(x + y)) + Float64(log(z) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -3.1e+18) || ~((a <= 1.6e+50)))
		tmp = log(t) * a;
	else
		tmp = log((x + y)) + (log(z) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -3.1e+18], N[Not[LessEqual[a, 1.6e+50]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.1e18 or 1.59999999999999991e50 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. expm1-log1p-u 45.6%

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

      2. expm1-undefine 45.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]

   4. Applied egg-rr 45.6%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-define 45.6%

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

   6. Simplified 45.6%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 45.6%

         \[\leadsto \left(\color{blue}{1 \cdot \left(\log \left(x + y\right) + \log z\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      2. +-commutative 45.6%

         \[\leadsto \left(1 \cdot \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      3. sum-log 35.3%

         \[\leadsto \left(1 \cdot \color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

   8. Applied egg-rr 35.3%

      \[\leadsto \left(\color{blue}{1 \cdot \log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]
   9. Step-by-step derivation

      1. *-lft-identity 35.3%

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

      2. +-commutative 35.3%

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

   10. Simplified 35.3%

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

   11. Taylor expanded in a around inf 81.3%

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

       1. *-commutative 81.3%

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

   13. Simplified 81.3%

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

   if -3.1e18 < a < 1.59999999999999991e50

   1. Initial program 99.5%

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

      1. associate-+l- 99.6%

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

      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 59.9%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{+18} \lor \neg \left(a \leq 1.6 \cdot 10^{+50}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{+164}:\\ \;\;\;\;\left(\log t \cdot \left(a - 0.5\right) + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.7e+164)
   (- (+ (* (log t) (- a 0.5)) (log (* y z))) t)
   (+ (- 1.0 t) -1.0)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.7e+164) {
		tmp = ((log(t) * (a - 0.5)) + log((y * z))) - t;
	} else {
		tmp = (1.0 - t) + -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 <= 1.7d+164) then
        tmp = ((log(t) * (a - 0.5d0)) + log((y * z))) - t
    else
        tmp = (1.0d0 - t) + (-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 <= 1.7e+164) {
		tmp = ((Math.log(t) * (a - 0.5)) + Math.log((y * z))) - t;
	} else {
		tmp = (1.0 - t) + -1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.7e+164:
		tmp = ((math.log(t) * (a - 0.5)) + math.log((y * z))) - t
	else:
		tmp = (1.0 - t) + -1.0
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.7e+164)
		tmp = Float64(Float64(Float64(log(t) * Float64(a - 0.5)) + log(Float64(y * z))) - t);
	else
		tmp = Float64(Float64(1.0 - t) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 1.7e+164)
		tmp = ((log(t) * (a - 0.5)) + log((y * z))) - t;
	else
		tmp = (1.0 - t) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.7e+164], N[(N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(1.0 - t), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.7000000000000001e164

   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. Add Preprocessing
   3. Step-by-step derivation

      1. expm1-log1p-u 32.6%

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

      2. expm1-undefine 32.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]

   4. Applied egg-rr 32.6%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-define 32.6%

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

   6. Simplified 32.6%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 32.6%

         \[\leadsto \left(\color{blue}{1 \cdot \left(\log \left(x + y\right) + \log z\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      2. +-commutative 32.6%

         \[\leadsto \left(1 \cdot \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      3. sum-log 28.1%

         \[\leadsto \left(1 \cdot \color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

   8. Applied egg-rr 28.1%

      \[\leadsto \left(\color{blue}{1 \cdot \log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]
   9. Step-by-step derivation

      1. *-lft-identity 28.1%

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

      2. +-commutative 28.1%

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

   10. Simplified 28.1%

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

   11. Taylor expanded in x around 0 57.3%

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

   if 1.7000000000000001e164 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. expm1-log1p-u 99.8%

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

      2. expm1-undefine 99.8%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-define 99.8%

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

   6. Simplified 99.8%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 99.8%

         \[\leadsto \left(\color{blue}{1 \cdot \left(\log \left(x + y\right) + \log z\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      2. +-commutative 99.8%

         \[\leadsto \left(1 \cdot \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      3. sum-log 67.5%

         \[\leadsto \left(1 \cdot \color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

   8. Applied egg-rr 67.5%

      \[\leadsto \left(\color{blue}{1 \cdot \log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]
   9. Step-by-step derivation

      1. *-lft-identity 67.5%

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

      2. +-commutative 67.5%

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

   10. Simplified 67.5%

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

   11. Taylor expanded in t around inf 87.8%

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

       1. neg-mul-1 87.8%

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

   13. Simplified 87.8%

       \[\leadsto \color{blue}{-t} \]
   14. 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%

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

   15. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]
   16. 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 87.8%

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

       4. unsub-neg 87.8%

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

       5. metadata-eval 87.8%

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

   17. Simplified 87.8%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{+164}:\\ \;\;\;\;\left(\log t \cdot \left(a - 0.5\right) + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 21000:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) + \log \left(\left(x + y\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 21000.0)
   (+ (* (log t) (+ a -0.5)) (log (* (+ x y) z)))
   (+ (log (+ x y)) (- (log z) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 21000.0) {
		tmp = (log(t) * (a + -0.5)) + log(((x + y) * z));
	} else {
		tmp = log((x + y)) + (log(z) - t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 21000.0d0) then
        tmp = (log(t) * (a + (-0.5d0))) + log(((x + y) * z))
    else
        tmp = log((x + y)) + (log(z) - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 21000.0) {
		tmp = (Math.log(t) * (a + -0.5)) + Math.log(((x + y) * z));
	} else {
		tmp = Math.log((x + y)) + (Math.log(z) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 21000.0:
		tmp = (math.log(t) * (a + -0.5)) + math.log(((x + y) * z))
	else:
		tmp = math.log((x + y)) + (math.log(z) - t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 21000.0)
		tmp = Float64(Float64(log(t) * Float64(a + -0.5)) + log(Float64(Float64(x + y) * z)));
	else
		tmp = Float64(log(Float64(x + y)) + Float64(log(z) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 21000.0)
		tmp = (log(t) * (a + -0.5)) + log(((x + y) * z));
	else
		tmp = log((x + y)) + (log(z) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 21000.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision] + N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 21000

   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. Add Preprocessing
   3. Step-by-step derivation

      1. expm1-log1p-u 0.8%

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

      2. expm1-undefine 0.8%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]

   4. Applied egg-rr 0.8%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-define 0.8%

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

   6. Simplified 0.8%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 0.8%

         \[\leadsto \left(\color{blue}{1 \cdot \left(\log \left(x + y\right) + \log z\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      2. +-commutative 0.8%

         \[\leadsto \left(1 \cdot \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      3. sum-log 0.8%

         \[\leadsto \left(1 \cdot \color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

   8. Applied egg-rr 0.8%

      \[\leadsto \left(\color{blue}{1 \cdot \log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]
   9. Step-by-step derivation

      1. *-lft-identity 0.8%

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

      2. +-commutative 0.8%

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

   10. Simplified 0.8%

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

   11. Taylor expanded in t around 0 75.0%

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

       1. +-commutative 75.0%

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

       2. sub-neg 75.0%

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

       3. metadata-eval 75.0%

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

       4. +-commutative 75.0%

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

   13. Simplified 75.0%

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

   if 21000 < 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 74.2%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 21000:\\ \;\;\;\;\log t \cdot \left(a + -0.5\right) + \log \left(\left(x + y\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \left(\log z - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.85 \cdot 10^{+21} \lor \neg \left(a \leq 1.85 \cdot 10^{+50}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.85e+21) (not (<= a 1.85e+50)))
   (* (log t) a)
   (- (+ (log z) (log y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.85e+21) || !(a <= 1.85e+50)) {
		tmp = log(t) * a;
	} else {
		tmp = (log(z) + 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 <= (-1.85d+21)) .or. (.not. (a <= 1.85d+50))) then
        tmp = log(t) * a
    else
        tmp = (log(z) + 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 <= -1.85e+21) || !(a <= 1.85e+50)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = (Math.log(z) + Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -1.85e+21) or not (a <= 1.85e+50):
		tmp = math.log(t) * a
	else:
		tmp = (math.log(z) + math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.85e+21) || !(a <= 1.85e+50))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(Float64(log(z) + log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -1.85e+21) || ~((a <= 1.85e+50)))
		tmp = log(t) * a;
	else
		tmp = (log(z) + log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.85e+21], N[Not[LessEqual[a, 1.85e+50]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.85e21 or 1.85e50 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. expm1-log1p-u 45.6%

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

      2. expm1-undefine 45.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]

   4. Applied egg-rr 45.6%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-define 45.6%

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

   6. Simplified 45.6%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 45.6%

         \[\leadsto \left(\color{blue}{1 \cdot \left(\log \left(x + y\right) + \log z\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      2. +-commutative 45.6%

         \[\leadsto \left(1 \cdot \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      3. sum-log 35.3%

         \[\leadsto \left(1 \cdot \color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

   8. Applied egg-rr 35.3%

      \[\leadsto \left(\color{blue}{1 \cdot \log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]
   9. Step-by-step derivation

      1. *-lft-identity 35.3%

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

      2. +-commutative 35.3%

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

   10. Simplified 35.3%

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

   11. Taylor expanded in a around inf 81.3%

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

       1. *-commutative 81.3%

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

   13. Simplified 81.3%

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

   if -1.85e21 < a < 1.85e50

   1. Initial program 99.5%

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

      1. associate-+l- 99.6%

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

      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 59.9%

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

   6. Taylor expanded in x around 0 39.3%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.85 \cdot 10^{+21} \lor \neg \left(a \leq 1.85 \cdot 10^{+50}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.15 \cdot 10^{+20} \lor \neg \left(a \leq 2.7 \cdot 10^{+50}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.15e+20) (not (<= a 2.7e+50))) (* (log t) a) (- (log y) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.15e+20) || !(a <= 2.7e+50)) {
		tmp = log(t) * a;
	} 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 <= (-1.15d+20)) .or. (.not. (a <= 2.7d+50))) then
        tmp = log(t) * a
    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 <= -1.15e+20) || !(a <= 2.7e+50)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = Math.log(y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -1.15e+20) or not (a <= 2.7e+50):
		tmp = math.log(t) * a
	else:
		tmp = math.log(y) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.15e+20) || !(a <= 2.7e+50))
		tmp = Float64(log(t) * a);
	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 <= -1.15e+20) || ~((a <= 2.7e+50)))
		tmp = log(t) * a;
	else
		tmp = log(y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.15e+20], N[Not[LessEqual[a, 2.7e+50]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.15e20 or 2.7e50 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. expm1-log1p-u 45.6%

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

      2. expm1-undefine 45.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]

   4. Applied egg-rr 45.6%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-define 45.6%

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

   6. Simplified 45.6%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 45.6%

         \[\leadsto \left(\color{blue}{1 \cdot \left(\log \left(x + y\right) + \log z\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      2. +-commutative 45.6%

         \[\leadsto \left(1 \cdot \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      3. sum-log 35.3%

         \[\leadsto \left(1 \cdot \color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

   8. Applied egg-rr 35.3%

      \[\leadsto \left(\color{blue}{1 \cdot \log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]
   9. Step-by-step derivation

      1. *-lft-identity 35.3%

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

      2. +-commutative 35.3%

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

   10. Simplified 35.3%

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

   11. Taylor expanded in a around inf 81.3%

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

       1. *-commutative 81.3%

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

   13. Simplified 81.3%

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

   if -1.15e20 < a < 2.7e50

   1. Initial program 99.5%

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

      1. associate-+l- 99.6%

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

      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 58.7%

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

      1. neg-mul-1 58.7%

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

   7. Simplified 58.7%

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

   8. Taylor expanded in x around 0 39.3%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.15 \cdot 10^{+20} \lor \neg \left(a \leq 2.7 \cdot 10^{+50}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{+19} \lor \neg \left(a \leq 1.72 \cdot 10^{+50}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -5.8e+19) (not (<= a 1.72e+50)))
   (* (log t) a)
   (+ (- 1.0 t) -1.0)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -5.8e+19) || !(a <= 1.72e+50)) {
		tmp = log(t) * a;
	} else {
		tmp = (1.0 - t) + -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 ((a <= (-5.8d+19)) .or. (.not. (a <= 1.72d+50))) then
        tmp = log(t) * a
    else
        tmp = (1.0d0 - t) + (-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 ((a <= -5.8e+19) || !(a <= 1.72e+50)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = (1.0 - t) + -1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -5.8e+19) or not (a <= 1.72e+50):
		tmp = math.log(t) * a
	else:
		tmp = (1.0 - t) + -1.0
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -5.8e+19) || !(a <= 1.72e+50))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(Float64(1.0 - t) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -5.8e+19) || ~((a <= 1.72e+50)))
		tmp = log(t) * a;
	else
		tmp = (1.0 - t) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -5.8e+19], N[Not[LessEqual[a, 1.72e+50]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(N[(1.0 - t), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.8e19 or 1.72e50 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. expm1-log1p-u 45.6%

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

      2. expm1-undefine 45.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]

   4. Applied egg-rr 45.6%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-define 45.6%

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

   6. Simplified 45.6%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 45.6%

         \[\leadsto \left(\color{blue}{1 \cdot \left(\log \left(x + y\right) + \log z\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      2. +-commutative 45.6%

         \[\leadsto \left(1 \cdot \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      3. sum-log 35.3%

         \[\leadsto \left(1 \cdot \color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

   8. Applied egg-rr 35.3%

      \[\leadsto \left(\color{blue}{1 \cdot \log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]
   9. Step-by-step derivation

      1. *-lft-identity 35.3%

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

      2. +-commutative 35.3%

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

   10. Simplified 35.3%

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

   11. Taylor expanded in a around inf 81.3%

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

       1. *-commutative 81.3%

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

   13. Simplified 81.3%

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

   if -5.8e19 < a < 1.72e50

   1. Initial program 99.5%

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

      1. expm1-log1p-u 52.4%

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

      2. expm1-undefine 52.4%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]

   4. Applied egg-rr 52.4%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-define 52.4%

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

   6. Simplified 52.4%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 52.4%

         \[\leadsto \left(\color{blue}{1 \cdot \left(\log \left(x + y\right) + \log z\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      2. +-commutative 52.4%

         \[\leadsto \left(1 \cdot \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      3. sum-log 40.1%

         \[\leadsto \left(1 \cdot \color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

   8. Applied egg-rr 40.1%

      \[\leadsto \left(\color{blue}{1 \cdot \log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]
   9. Step-by-step derivation

      1. *-lft-identity 40.1%

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

      2. +-commutative 40.1%

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

   10. Simplified 40.1%

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

   11. Taylor expanded in t around inf 52.8%

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

       1. neg-mul-1 52.8%

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

   13. Simplified 52.8%

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

       1. expm1-log1p-u 1.5%

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

       2. expm1-undefine 1.6%

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

   15. Applied egg-rr 1.6%

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

       1. sub-neg 1.6%

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

       2. log1p-undefine 1.6%

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

       3. rem-exp-log 52.9%

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

       4. unsub-neg 52.9%

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

       5. metadata-eval 52.9%

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

   17. Simplified 52.9%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{+19} \lor \neg \left(a \leq 1.72 \cdot 10^{+50}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 39.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 600:\\ \;\;\;\;\log y\\ \mathbf{else}:\\ \;\;\;\;\left(1 - t\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 600.0) (log y) (+ (- 1.0 t) -1.0)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 600.0) {
		tmp = log(y);
	} else {
		tmp = (1.0 - t) + -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 <= 600.0d0) then
        tmp = log(y)
    else
        tmp = (1.0d0 - t) + (-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 <= 600.0) {
		tmp = Math.log(y);
	} else {
		tmp = (1.0 - t) + -1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 600.0:
		tmp = math.log(y)
	else:
		tmp = (1.0 - t) + -1.0
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 600.0)
		tmp = log(y);
	else
		tmp = Float64(Float64(1.0 - t) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 600.0)
		tmp = log(y);
	else
		tmp = (1.0 - t) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 600.0], N[Log[y], $MachinePrecision], N[(N[(1.0 - t), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 600

   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 9.8%

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

      1. neg-mul-1 9.8%

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

   7. Simplified 9.8%

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

   8. Taylor expanded in t around 0 9.8%

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

      1. +-commutative 9.8%

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

   10. Simplified 9.8%

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

   11. Taylor expanded in y around inf 7.2%

       \[\leadsto \color{blue}{-1 \cdot \log \left(\frac{1}{y}\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg 7.2%

          \[\leadsto \color{blue}{-\log \left(\frac{1}{y}\right)} \]

       2. log-rec 7.2%

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

       3. remove-double-neg 7.2%

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

   13. Simplified 7.2%

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

   if 600 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. expm1-log1p-u 99.6%

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

      2. expm1-undefine 99.6%

         \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]

   4. Applied egg-rr 99.6%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-define 99.6%

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

   6. Simplified 99.6%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 99.6%

         \[\leadsto \left(\color{blue}{1 \cdot \left(\log \left(x + y\right) + \log z\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      2. +-commutative 99.6%

         \[\leadsto \left(1 \cdot \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

      3. sum-log 76.5%

         \[\leadsto \left(1 \cdot \color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

   8. Applied egg-rr 76.5%

      \[\leadsto \left(\color{blue}{1 \cdot \log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]
   9. Step-by-step derivation

      1. *-lft-identity 76.5%

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

      2. +-commutative 76.5%

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

   10. Simplified 76.5%

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

   11. Taylor expanded in t around inf 73.5%

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

       1. neg-mul-1 73.5%

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

   13. Simplified 73.5%

       \[\leadsto \color{blue}{-t} \]
   14. 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%

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

   15. Applied egg-rr 0.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]
   16. 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 73.5%

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

       4. unsub-neg 73.5%

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

       5. metadata-eval 73.5%

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

   17. Simplified 73.5%

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

Alternative 16: 37.8% accurate, 62.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return (1.0 - t) + -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 = (1.0d0 - t) + (-1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(1.0 - t), $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. Add Preprocessing
3. Step-by-step derivation

   1. expm1-log1p-u 49.4%

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

   2. expm1-undefine 49.4%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]

4. Applied egg-rr 49.4%

   \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]
5. Step-by-step derivation

   1. expm1-define 49.4%

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

6. Simplified 49.4%

   \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right)} \]
7. Step-by-step derivation

   1. *-un-lft-identity 49.4%

      \[\leadsto \left(\color{blue}{1 \cdot \left(\log \left(x + y\right) + \log z\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

   2. +-commutative 49.4%

      \[\leadsto \left(1 \cdot \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

   3. sum-log 38.0%

      \[\leadsto \left(1 \cdot \color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

8. Applied egg-rr 38.0%

   \[\leadsto \left(\color{blue}{1 \cdot \log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]
9. Step-by-step derivation

   1. *-lft-identity 38.0%

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

   2. +-commutative 38.0%

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

10. Simplified 38.0%

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

11. Taylor expanded in t around inf 37.9%

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

    1. neg-mul-1 37.9%

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

13. Simplified 37.9%

    \[\leadsto \color{blue}{-t} \]
14. 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%

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

15. Applied egg-rr 1.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]
16. 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 37.9%

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

    4. unsub-neg 37.9%

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

    5. metadata-eval 37.9%

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

17. Simplified 37.9%

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

Alternative 17: 37.8% accurate, 156.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. expm1-log1p-u 49.4%

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

   2. expm1-undefine 49.4%

      \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]

4. Applied egg-rr 49.4%

   \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log t\right)} - 1\right)} \]
5. Step-by-step derivation

   1. expm1-define 49.4%

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

6. Simplified 49.4%

   \[\leadsto \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right)} \]
7. Step-by-step derivation

   1. *-un-lft-identity 49.4%

      \[\leadsto \left(\color{blue}{1 \cdot \left(\log \left(x + y\right) + \log z\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

   2. +-commutative 49.4%

      \[\leadsto \left(1 \cdot \color{blue}{\left(\log z + \log \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

   3. sum-log 38.0%

      \[\leadsto \left(1 \cdot \color{blue}{\log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]

8. Applied egg-rr 38.0%

   \[\leadsto \left(\color{blue}{1 \cdot \log \left(z \cdot \left(x + y\right)\right)} - t\right) + \left(a - 0.5\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\log t\right)\right) \]
9. Step-by-step derivation

   1. *-lft-identity 38.0%

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

   2. +-commutative 38.0%

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

10. Simplified 38.0%

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

11. Taylor expanded in t around inf 37.9%

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

    1. neg-mul-1 37.9%

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

13. Simplified 37.9%

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

Developer target: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024110 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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