Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A

Percentage Accurate: 99.9% → 99.9%

Time: 11.2s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative N/A

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

   2. associate--l- N/A

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

   3. associate-+r- N/A

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

   4. +-commutative N/A

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

   5. associate--r+ N/A

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

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(\log t + x \cdot \log y\right) - z\right), \color{blue}{y}\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\log t + x \cdot \log y\right), z\right), y\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\log t, \left(x \cdot \log y\right)\right), z\right), y\right) \]

   9. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \left(x \cdot \log y\right)\right), z\right), y\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{*.f64}\left(x, \log y\right)\right), z\right), y\right) \]

   11. log-lowering-log.f64 99.9%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right)\right), z\right), y\right) \]

4. Applied egg-rr 99.9%

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

Alternative 2: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - y\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+135}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq -2000:\\ \;\;\;\;0 - \left(y + z\right)\\ \mathbf{elif}\;t\_1 \leq 10^{-12}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) y)))
   (if (<= t_1 -1e+135)
     t_1
     (if (<= t_1 -2000.0)
       (- 0.0 (+ y z))
       (if (<= t_1 1e-12) (- (log t) z) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - y;
	double tmp;
	if (t_1 <= -1e+135) {
		tmp = t_1;
	} else if (t_1 <= -2000.0) {
		tmp = 0.0 - (y + z);
	} else if (t_1 <= 1e-12) {
		tmp = log(t) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * log(y)) - y
    if (t_1 <= (-1d+135)) then
        tmp = t_1
    else if (t_1 <= (-2000.0d0)) then
        tmp = 0.0d0 - (y + z)
    else if (t_1 <= 1d-12) then
        tmp = log(t) - z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * Math.log(y)) - y;
	double tmp;
	if (t_1 <= -1e+135) {
		tmp = t_1;
	} else if (t_1 <= -2000.0) {
		tmp = 0.0 - (y + z);
	} else if (t_1 <= 1e-12) {
		tmp = Math.log(t) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - y
	tmp = 0
	if t_1 <= -1e+135:
		tmp = t_1
	elif t_1 <= -2000.0:
		tmp = 0.0 - (y + z)
	elif t_1 <= 1e-12:
		tmp = math.log(t) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - y)
	tmp = 0.0
	if (t_1 <= -1e+135)
		tmp = t_1;
	elseif (t_1 <= -2000.0)
		tmp = Float64(0.0 - Float64(y + z));
	elseif (t_1 <= 1e-12)
		tmp = Float64(log(t) - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * log(y)) - y;
	tmp = 0.0;
	if (t_1 <= -1e+135)
		tmp = t_1;
	elseif (t_1 <= -2000.0)
		tmp = 0.0 - (y + z);
	elseif (t_1 <= 1e-12)
		tmp = log(t) - z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+135], t$95$1, If[LessEqual[t$95$1, -2000.0], N[(0.0 - N[(y + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-12], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -9.99999999999999962e134 or 9.9999999999999998e-13 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.8%

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

      1. +-commutative N/A

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

      2. associate--l- N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

      5. associate--r+ N/A

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

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\log t + x \cdot \log y\right) - z\right), \color{blue}{y}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\log t + x \cdot \log y\right), z\right), y\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\log t, \left(x \cdot \log y\right)\right), z\right), y\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \left(x \cdot \log y\right)\right), z\right), y\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{*.f64}\left(x, \log y\right)\right), z\right), y\right) \]

      11. log-lowering-log.f64 99.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right)\right), z\right), y\right) \]

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(x \cdot \log y\right)}, y\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log y\right), y\right) \]

      2. log-lowering-log.f64 86.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), y\right) \]

   7. Simplified 86.7%

      \[\leadsto \color{blue}{x \cdot \log y} - y \]

   if -9.99999999999999962e134 < (-.f64 (*.f64 x (log.f64 y)) y) < -2e3

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. associate--l- N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

      5. associate--r+ N/A

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

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\log t + x \cdot \log y\right) - z\right), \color{blue}{y}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\log t + x \cdot \log y\right), z\right), y\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\log t, \left(x \cdot \log y\right)\right), z\right), y\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \left(x \cdot \log y\right)\right), z\right), y\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{*.f64}\left(x, \log y\right)\right), z\right), y\right) \]

      11. log-lowering-log.f64 99.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right)\right), z\right), y\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot z\right)}, y\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), y\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(0 - z\right), y\right) \]

      3. --lowering--.f64 79.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), y\right) \]

   7. Simplified 79.3%

      \[\leadsto \color{blue}{\left(0 - z\right)} - y \]

   if -2e3 < (-.f64 (*.f64 x (log.f64 y)) y) < 9.9999999999999998e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t + \left(\mathsf{neg}\left(z\right)\right)\right), \color{blue}{y}\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t - z\right), y\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\log t, z\right), y\right) \]

      7. log-lowering-log.f64 99.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(t\right), z\right), y\right) \]

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\log t, \color{blue}{z}\right) \]

      2. log-lowering-log.f64 99.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log.f64}\left(t\right), z\right) \]

   8. Simplified 99.5%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -1 \cdot 10^{+135}:\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{elif}\;x \cdot \log y - y \leq -2000:\\ \;\;\;\;0 - \left(y + z\right)\\ \mathbf{elif}\;x \cdot \log y - y \leq 10^{-12}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - y\\ t_3 := \left(t\_1 - z\right) - y\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+26}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{-12}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 y)) (t_3 (- (- t_1 z) y)))
   (if (<= t_2 -5e+26) t_3 (if (<= t_2 1e-12) (- (- (log t) z) y) t_3))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = t_1 - y;
	double t_3 = (t_1 - z) - y;
	double tmp;
	if (t_2 <= -5e+26) {
		tmp = t_3;
	} else if (t_2 <= 1e-12) {
		tmp = (log(t) - z) - y;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = t_1 - y
    t_3 = (t_1 - z) - y
    if (t_2 <= (-5d+26)) then
        tmp = t_3
    else if (t_2 <= 1d-12) then
        tmp = (log(t) - z) - y
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double t_2 = t_1 - y;
	double t_3 = (t_1 - z) - y;
	double tmp;
	if (t_2 <= -5e+26) {
		tmp = t_3;
	} else if (t_2 <= 1e-12) {
		tmp = (Math.log(t) - z) - y;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = t_1 - y
	t_3 = (t_1 - z) - y
	tmp = 0
	if t_2 <= -5e+26:
		tmp = t_3
	elif t_2 <= 1e-12:
		tmp = (math.log(t) - z) - y
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(t_1 - y)
	t_3 = Float64(Float64(t_1 - z) - y)
	tmp = 0.0
	if (t_2 <= -5e+26)
		tmp = t_3;
	elseif (t_2 <= 1e-12)
		tmp = Float64(Float64(log(t) - z) - y);
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = t_1 - y;
	t_3 = (t_1 - z) - y;
	tmp = 0.0;
	if (t_2 <= -5e+26)
		tmp = t_3;
	elseif (t_2 <= 1e-12)
		tmp = (log(t) - z) - y;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - y), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 - z), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+26], t$95$3, If[LessEqual[t$95$2, 1e-12], N[(N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision] - y), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -5.0000000000000001e26 or 9.9999999999999998e-13 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.8%

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

      1. +-commutative N/A

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

      2. associate--l- N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

      5. associate--r+ N/A

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

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\log t + x \cdot \log y\right) - z\right), \color{blue}{y}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\log t + x \cdot \log y\right), z\right), y\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\log t, \left(x \cdot \log y\right)\right), z\right), y\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \left(x \cdot \log y\right)\right), z\right), y\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{*.f64}\left(x, \log y\right)\right), z\right), y\right) \]

      11. log-lowering-log.f64 99.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right)\right), z\right), y\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(x \cdot \log y\right)}, z\right), y\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log y\right), z\right), y\right) \]

      2. log-lowering-log.f64 99.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), z\right), y\right) \]

   7. Simplified 99.4%

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

   if -5.0000000000000001e26 < (-.f64 (*.f64 x (log.f64 y)) y) < 9.9999999999999998e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t + \left(\mathsf{neg}\left(z\right)\right)\right), \color{blue}{y}\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t - z\right), y\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\log t, z\right), y\right) \]

      7. log-lowering-log.f64 99.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(t\right), z\right), y\right) \]

   5. Simplified 99.9%

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

Alternative 4: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - z\\ \mathbf{if}\;t\_1 - y \leq -2000:\\ \;\;\;\;t\_2 - y\\ \mathbf{else}:\\ \;\;\;\;\log t + t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 z)))
   (if (<= (- t_1 y) -2000.0) (- t_2 y) (+ (log t) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = t_1 - z;
	double tmp;
	if ((t_1 - y) <= -2000.0) {
		tmp = t_2 - y;
	} else {
		tmp = log(t) + t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = t_1 - z
    if ((t_1 - y) <= (-2000.0d0)) then
        tmp = t_2 - y
    else
        tmp = log(t) + t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double t_2 = t_1 - z;
	double tmp;
	if ((t_1 - y) <= -2000.0) {
		tmp = t_2 - y;
	} else {
		tmp = Math.log(t) + t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = t_1 - z
	tmp = 0
	if (t_1 - y) <= -2000.0:
		tmp = t_2 - y
	else:
		tmp = math.log(t) + t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(t_1 - z)
	tmp = 0.0
	if (Float64(t_1 - y) <= -2000.0)
		tmp = Float64(t_2 - y);
	else
		tmp = Float64(log(t) + t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = t_1 - z;
	tmp = 0.0;
	if ((t_1 - y) <= -2000.0)
		tmp = t_2 - y;
	else
		tmp = log(t) + t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - z), $MachinePrecision]}, If[LessEqual[N[(t$95$1 - y), $MachinePrecision], -2000.0], N[(t$95$2 - y), $MachinePrecision], N[(N[Log[t], $MachinePrecision] + t$95$2), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -2e3

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. associate--l- N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

      5. associate--r+ N/A

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

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\log t + x \cdot \log y\right) - z\right), \color{blue}{y}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\log t + x \cdot \log y\right), z\right), y\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\log t, \left(x \cdot \log y\right)\right), z\right), y\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \left(x \cdot \log y\right)\right), z\right), y\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{*.f64}\left(x, \log y\right)\right), z\right), y\right) \]

      11. log-lowering-log.f64 99.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right)\right), z\right), y\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(x \cdot \log y\right)}, z\right), y\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log y\right), z\right), y\right) \]

      2. log-lowering-log.f64 98.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), z\right), y\right) \]

   7. Simplified 98.6%

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

   if -2e3 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(x \cdot \log y - z\right)}, \mathsf{log.f64}\left(t\right)\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot \log y\right), z\right), \mathsf{log.f64}\left(\color{blue}{t}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log y\right), z\right), \mathsf{log.f64}\left(t\right)\right) \]

      3. log-lowering-log.f64 99.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), z\right), \mathsf{log.f64}\left(t\right)\right) \]

   5. Simplified 99.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -2000:\\ \;\;\;\;\left(x \cdot \log y - z\right) - y\\ \mathbf{else}:\\ \;\;\;\;\log t + \left(x \cdot \log y - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \left(t\_1 - z\right) - y\\ \mathbf{if}\;z \leq -17000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-8}:\\ \;\;\;\;\log t + \left(t\_1 - y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- (- t_1 z) y)))
   (if (<= z -17000000000.0)
     t_2
     (if (<= z 2.5e-8) (+ (log t) (- t_1 y)) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = (t_1 - z) - y;
	double tmp;
	if (z <= -17000000000.0) {
		tmp = t_2;
	} else if (z <= 2.5e-8) {
		tmp = log(t) + (t_1 - y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = (t_1 - z) - y
    if (z <= (-17000000000.0d0)) then
        tmp = t_2
    else if (z <= 2.5d-8) then
        tmp = log(t) + (t_1 - y)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double t_2 = (t_1 - z) - y;
	double tmp;
	if (z <= -17000000000.0) {
		tmp = t_2;
	} else if (z <= 2.5e-8) {
		tmp = Math.log(t) + (t_1 - y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = (t_1 - z) - y
	tmp = 0
	if z <= -17000000000.0:
		tmp = t_2
	elif z <= 2.5e-8:
		tmp = math.log(t) + (t_1 - y)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(Float64(t_1 - z) - y)
	tmp = 0.0
	if (z <= -17000000000.0)
		tmp = t_2;
	elseif (z <= 2.5e-8)
		tmp = Float64(log(t) + Float64(t_1 - y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = (t_1 - z) - y;
	tmp = 0.0;
	if (z <= -17000000000.0)
		tmp = t_2;
	elseif (z <= 2.5e-8)
		tmp = log(t) + (t_1 - y);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 - z), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[z, -17000000000.0], t$95$2, If[LessEqual[z, 2.5e-8], N[(N[Log[t], $MachinePrecision] + N[(t$95$1 - y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7e10 or 2.4999999999999999e-8 < z

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. associate--l- N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

      5. associate--r+ N/A

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

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\log t + x \cdot \log y\right) - z\right), \color{blue}{y}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\log t + x \cdot \log y\right), z\right), y\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\log t, \left(x \cdot \log y\right)\right), z\right), y\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \left(x \cdot \log y\right)\right), z\right), y\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{*.f64}\left(x, \log y\right)\right), z\right), y\right) \]

      11. log-lowering-log.f64 99.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right)\right), z\right), y\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(x \cdot \log y\right)}, z\right), y\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log y\right), z\right), y\right) \]

      2. log-lowering-log.f64 99.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), z\right), y\right) \]

   7. Simplified 99.4%

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

   if -1.7e10 < z < 2.4999999999999999e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(x \cdot \log y - y\right)}, \mathsf{log.f64}\left(t\right)\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot \log y\right), y\right), \mathsf{log.f64}\left(\color{blue}{t}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log y\right), y\right), \mathsf{log.f64}\left(t\right)\right) \]

      3. log-lowering-log.f64 99.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), y\right), \mathsf{log.f64}\left(t\right)\right) \]

   5. Simplified 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -17000000000:\\ \;\;\;\;\left(x \cdot \log y - z\right) - y\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-8}:\\ \;\;\;\;\log t + \left(x \cdot \log y - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log y - z\right) - y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 7: 69.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := 0 - \left(y + z\right)\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{+124}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-136}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -1.32 \cdot 10^{-238}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+38}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- 0.0 (+ y z))))
   (if (<= x -2.3e+124)
     t_1
     (if (<= x -3.4e-136)
       t_2
       (if (<= x -1.32e-238) (- (log t) y) (if (<= x 2.7e+38) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = 0.0 - (y + z);
	double tmp;
	if (x <= -2.3e+124) {
		tmp = t_1;
	} else if (x <= -3.4e-136) {
		tmp = t_2;
	} else if (x <= -1.32e-238) {
		tmp = log(t) - y;
	} else if (x <= 2.7e+38) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = 0.0d0 - (y + z)
    if (x <= (-2.3d+124)) then
        tmp = t_1
    else if (x <= (-3.4d-136)) then
        tmp = t_2
    else if (x <= (-1.32d-238)) then
        tmp = log(t) - y
    else if (x <= 2.7d+38) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double t_2 = 0.0 - (y + z);
	double tmp;
	if (x <= -2.3e+124) {
		tmp = t_1;
	} else if (x <= -3.4e-136) {
		tmp = t_2;
	} else if (x <= -1.32e-238) {
		tmp = Math.log(t) - y;
	} else if (x <= 2.7e+38) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = 0.0 - (y + z)
	tmp = 0
	if x <= -2.3e+124:
		tmp = t_1
	elif x <= -3.4e-136:
		tmp = t_2
	elif x <= -1.32e-238:
		tmp = math.log(t) - y
	elif x <= 2.7e+38:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(0.0 - Float64(y + z))
	tmp = 0.0
	if (x <= -2.3e+124)
		tmp = t_1;
	elseif (x <= -3.4e-136)
		tmp = t_2;
	elseif (x <= -1.32e-238)
		tmp = Float64(log(t) - y);
	elseif (x <= 2.7e+38)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = 0.0 - (y + z);
	tmp = 0.0;
	if (x <= -2.3e+124)
		tmp = t_1;
	elseif (x <= -3.4e-136)
		tmp = t_2;
	elseif (x <= -1.32e-238)
		tmp = log(t) - y;
	elseif (x <= 2.7e+38)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.0 - N[(y + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.3e+124], t$95$1, If[LessEqual[x, -3.4e-136], t$95$2, If[LessEqual[x, -1.32e-238], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision], If[LessEqual[x, 2.7e+38], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.29999999999999985e124 or 2.69999999999999996e38 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\log y}\right) \]

      2. log-lowering-log.f64 71.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right) \]

   5. Simplified 71.7%

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

   if -2.29999999999999985e124 < x < -3.4e-136 or -1.31999999999999998e-238 < x < 2.69999999999999996e38

   1. Initial program 100.0%

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

      1. +-commutative N/A

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

      2. associate--l- N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

      5. associate--r+ N/A

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

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\log t + x \cdot \log y\right) - z\right), \color{blue}{y}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\log t + x \cdot \log y\right), z\right), y\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\log t, \left(x \cdot \log y\right)\right), z\right), y\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \left(x \cdot \log y\right)\right), z\right), y\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{*.f64}\left(x, \log y\right)\right), z\right), y\right) \]

      11. log-lowering-log.f64 100.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right)\right), z\right), y\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot z\right)}, y\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), y\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(0 - z\right), y\right) \]

      3. --lowering--.f64 78.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), y\right) \]

   7. Simplified 78.5%

      \[\leadsto \color{blue}{\left(0 - z\right)} - y \]

   if -3.4e-136 < x < -1.31999999999999998e-238

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t + \left(\mathsf{neg}\left(z\right)\right)\right), \color{blue}{y}\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t - z\right), y\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\log t, z\right), y\right) \]

      7. log-lowering-log.f64 100.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(t\right), z\right), y\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in z around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\log t, \color{blue}{y}\right) \]

      2. log-lowering-log.f64 90.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log.f64}\left(t\right), y\right) \]

   8. Simplified 90.9%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+124}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-136}:\\ \;\;\;\;0 - \left(y + z\right)\\ \mathbf{elif}\;x \leq -1.32 \cdot 10^{-238}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+38}:\\ \;\;\;\;0 - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - y\\ \mathbf{if}\;x \leq -28:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+24}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) y)))
   (if (<= x -28.0) t_1 (if (<= x 1.5e+24) (- (- (log t) z) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - y;
	double tmp;
	if (x <= -28.0) {
		tmp = t_1;
	} else if (x <= 1.5e+24) {
		tmp = (log(t) - z) - y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * log(y)) - y
    if (x <= (-28.0d0)) then
        tmp = t_1
    else if (x <= 1.5d+24) then
        tmp = (log(t) - z) - y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * Math.log(y)) - y;
	double tmp;
	if (x <= -28.0) {
		tmp = t_1;
	} else if (x <= 1.5e+24) {
		tmp = (Math.log(t) - z) - y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - y
	tmp = 0
	if x <= -28.0:
		tmp = t_1
	elif x <= 1.5e+24:
		tmp = (math.log(t) - z) - y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - y)
	tmp = 0.0
	if (x <= -28.0)
		tmp = t_1;
	elseif (x <= 1.5e+24)
		tmp = Float64(Float64(log(t) - z) - y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * log(y)) - y;
	tmp = 0.0;
	if (x <= -28.0)
		tmp = t_1;
	elseif (x <= 1.5e+24)
		tmp = (log(t) - z) - y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[x, -28.0], t$95$1, If[LessEqual[x, 1.5e+24], N[(N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision] - y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -28 or 1.49999999999999997e24 < x

   1. Initial program 99.8%

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

      1. +-commutative N/A

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

      2. associate--l- N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

      5. associate--r+ N/A

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

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\log t + x \cdot \log y\right) - z\right), \color{blue}{y}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\log t + x \cdot \log y\right), z\right), y\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\log t, \left(x \cdot \log y\right)\right), z\right), y\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \left(x \cdot \log y\right)\right), z\right), y\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{*.f64}\left(x, \log y\right)\right), z\right), y\right) \]

      11. log-lowering-log.f64 99.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right)\right), z\right), y\right) \]

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(x \cdot \log y\right)}, y\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log y\right), y\right) \]

      2. log-lowering-log.f64 83.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), y\right) \]

   7. Simplified 83.9%

      \[\leadsto \color{blue}{x \cdot \log y} - y \]

   if -28 < x < 1.49999999999999997e24

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t + \left(\mathsf{neg}\left(z\right)\right)\right), \color{blue}{y}\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t - z\right), y\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\log t, z\right), y\right) \]

      7. log-lowering-log.f64 100.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(t\right), z\right), y\right) \]

   5. Simplified 100.0%

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

Alternative 9: 70.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{+124}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+38}:\\ \;\;\;\;0 - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -5.5e+124) t_1 (if (<= x 3.3e+38) (- 0.0 (+ y z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -5.5e+124) {
		tmp = t_1;
	} else if (x <= 3.3e+38) {
		tmp = 0.0 - (y + z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (x <= (-5.5d+124)) then
        tmp = t_1
    else if (x <= 3.3d+38) then
        tmp = 0.0d0 - (y + z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if (x <= -5.5e+124) {
		tmp = t_1;
	} else if (x <= 3.3e+38) {
		tmp = 0.0 - (y + z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -5.5e+124:
		tmp = t_1
	elif x <= 3.3e+38:
		tmp = 0.0 - (y + z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -5.5e+124)
		tmp = t_1;
	elseif (x <= 3.3e+38)
		tmp = Float64(0.0 - Float64(y + z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -5.5e+124)
		tmp = t_1;
	elseif (x <= 3.3e+38)
		tmp = 0.0 - (y + z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.5e+124], t$95$1, If[LessEqual[x, 3.3e+38], N[(0.0 - N[(y + z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.49999999999999977e124 or 3.2999999999999999e38 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\log y}\right) \]

      2. log-lowering-log.f64 71.7%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right) \]

   5. Simplified 71.7%

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

   if -5.49999999999999977e124 < x < 3.2999999999999999e38

   1. Initial program 100.0%

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

      1. +-commutative N/A

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

      2. associate--l- N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

      5. associate--r+ N/A

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

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\log t + x \cdot \log y\right) - z\right), \color{blue}{y}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\log t + x \cdot \log y\right), z\right), y\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\log t, \left(x \cdot \log y\right)\right), z\right), y\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \left(x \cdot \log y\right)\right), z\right), y\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{*.f64}\left(x, \log y\right)\right), z\right), y\right) \]

      11. log-lowering-log.f64 100.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right)\right), z\right), y\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot z\right)}, y\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), y\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(0 - z\right), y\right) \]

      3. --lowering--.f64 74.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), y\right) \]

   7. Simplified 74.3%

      \[\leadsto \color{blue}{\left(0 - z\right)} - y \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+124}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+38}:\\ \;\;\;\;0 - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-116}:\\ \;\;\;\;\log t\\ \mathbf{else}:\\ \;\;\;\;0 - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.3e-116) (log t) (- 0.0 (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.3e-116) {
		tmp = log(t);
	} else {
		tmp = 0.0 - (y + z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 2.3d-116) then
        tmp = log(t)
    else
        tmp = 0.0d0 - (y + z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.3e-116) {
		tmp = Math.log(t);
	} else {
		tmp = 0.0 - (y + z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 2.3e-116:
		tmp = math.log(t)
	else:
		tmp = 0.0 - (y + z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.3e-116)
		tmp = log(t);
	else
		tmp = Float64(0.0 - Float64(y + z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.3e-116)
		tmp = log(t);
	else
		tmp = 0.0 - (y + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.3e-116], N[Log[t], $MachinePrecision], N[(0.0 - N[(y + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.30000000000000002e-116

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t + \left(\mathsf{neg}\left(z\right)\right)\right), \color{blue}{y}\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log t - z\right), y\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\log t, z\right), y\right) \]

      7. log-lowering-log.f64 63.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{log.f64}\left(t\right), z\right), y\right) \]

   5. Simplified 63.7%

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

   6. Taylor expanded in z around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\log t, \color{blue}{y}\right) \]

      2. log-lowering-log.f64 37.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log.f64}\left(t\right), y\right) \]

   8. Simplified 37.9%

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

   9. Taylor expanded in y around 0

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

       1. log-lowering-log.f64 37.9%

          \[\leadsto \mathsf{log.f64}\left(t\right) \]

   11. Simplified 37.9%

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

   if 2.30000000000000002e-116 < y

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. associate--l- N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

      5. associate--r+ N/A

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

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\log t + x \cdot \log y\right) - z\right), \color{blue}{y}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\log t + x \cdot \log y\right), z\right), y\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\log t, \left(x \cdot \log y\right)\right), z\right), y\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \left(x \cdot \log y\right)\right), z\right), y\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{*.f64}\left(x, \log y\right)\right), z\right), y\right) \]

      11. log-lowering-log.f64 99.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right)\right), z\right), y\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot z\right)}, y\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), y\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(0 - z\right), y\right) \]

      3. --lowering--.f64 69.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), y\right) \]

   7. Simplified 69.3%

      \[\leadsto \color{blue}{\left(0 - z\right)} - y \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-116}:\\ \;\;\;\;\log t\\ \mathbf{else}:\\ \;\;\;\;0 - \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.1% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -460000000000:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+117}:\\ \;\;\;\;0 - y\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -460000000000.0)
   (- 0.0 z)
   (if (<= z 1.9e+117) (- 0.0 y) (- 0.0 z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -460000000000.0) {
		tmp = 0.0 - z;
	} else if (z <= 1.9e+117) {
		tmp = 0.0 - y;
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-460000000000.0d0)) then
        tmp = 0.0d0 - z
    else if (z <= 1.9d+117) then
        tmp = 0.0d0 - y
    else
        tmp = 0.0d0 - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -460000000000.0) {
		tmp = 0.0 - z;
	} else if (z <= 1.9e+117) {
		tmp = 0.0 - y;
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -460000000000.0:
		tmp = 0.0 - z
	elif z <= 1.9e+117:
		tmp = 0.0 - y
	else:
		tmp = 0.0 - z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -460000000000.0)
		tmp = Float64(0.0 - z);
	elseif (z <= 1.9e+117)
		tmp = Float64(0.0 - y);
	else
		tmp = Float64(0.0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -460000000000.0)
		tmp = 0.0 - z;
	elseif (z <= 1.9e+117)
		tmp = 0.0 - y;
	else
		tmp = 0.0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -460000000000.0], N[(0.0 - z), $MachinePrecision], If[LessEqual[z, 1.9e+117], N[(0.0 - y), $MachinePrecision], N[(0.0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.6e11 or 1.9000000000000001e117 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(z\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{z} \]

      3. --lowering--.f64 60.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

   5. Simplified 60.8%

      \[\leadsto \color{blue}{0 - z} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(z\right) \]

      2. neg-lowering-neg.f64 60.8%

         \[\leadsto \mathsf{neg.f64}\left(z\right) \]

   7. Applied egg-rr 60.8%

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

   if -4.6e11 < z < 1.9000000000000001e117

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(y\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{y} \]

      3. --lowering--.f64 44.0%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{y}\right) \]

   5. Simplified 44.0%

      \[\leadsto \color{blue}{0 - y} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(y\right) \]

      2. neg-lowering-neg.f64 44.0%

         \[\leadsto \mathsf{neg.f64}\left(y\right) \]

   7. Applied egg-rr 44.0%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -460000000000:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+117}:\\ \;\;\;\;0 - y\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.3% accurate, 41.8× speedup

Math

\[\begin{array}{l} \\ 0 - \left(y + z\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- 0.0 (+ y z)))

C

double code(double x, double y, double z, double t) {
	return 0.0 - (y + z);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 0.0d0 - (y + z)
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.0 - (y + z);
}

Python

def code(x, y, z, t):
	return 0.0 - (y + z)

Julia

function code(x, y, z, t)
	return Float64(0.0 - Float64(y + z))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.0 - (y + z);
end

Wolfram

code[x_, y_, z_, t_] := N[(0.0 - N[(y + z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. +-commutative N/A

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

   2. associate--l- N/A

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

   3. associate-+r- N/A

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

   4. +-commutative N/A

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

   5. associate--r+ N/A

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

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(\log t + x \cdot \log y\right) - z\right), \color{blue}{y}\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\log t + x \cdot \log y\right), z\right), y\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\log t, \left(x \cdot \log y\right)\right), z\right), y\right) \]

   9. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \left(x \cdot \log y\right)\right), z\right), y\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{*.f64}\left(x, \log y\right)\right), z\right), y\right) \]

   11. log-lowering-log.f64 99.9%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(t\right), \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right)\right), z\right), y\right) \]

4. Applied egg-rr 99.9%

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

5. Taylor expanded in z around inf

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot z\right)}, y\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(z\right)\right), y\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(0 - z\right), y\right) \]

   3. --lowering--.f64 57.7%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, z\right), y\right) \]

7. Simplified 57.7%

   \[\leadsto \color{blue}{\left(0 - z\right)} - y \]

8. Final simplification 57.7%

   \[\leadsto 0 - \left(y + z\right) \]
9. Add Preprocessing

Alternative 13: 30.1% accurate, 69.7× speedup

Math

\[\begin{array}{l} \\ 0 - y \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- 0.0 y))

C

double code(double x, double y, double z, double t) {
	return 0.0 - y;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 0.0d0 - y
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.0 - y;
}

Python

def code(x, y, z, t):
	return 0.0 - y

Julia

function code(x, y, z, t)
	return Float64(0.0 - y)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.0 - y;
end

Wolfram

code[x_, y_, z_, t_] := N[(0.0 - y), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(y\right) \]

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{y} \]

   3. --lowering--.f64 34.1%

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{y}\right) \]

5. Simplified 34.1%

   \[\leadsto \color{blue}{0 - y} \]
6. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{neg}\left(y\right) \]

   2. neg-lowering-neg.f64 34.1%

      \[\leadsto \mathsf{neg.f64}\left(y\right) \]

7. Applied egg-rr 34.1%

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

8. Final simplification 34.1%

   \[\leadsto 0 - y \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024161 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (- (* x (log y)) y) z) (log t)))