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

Percentage Accurate: 99.9% → 99.9%

Time: 10.2s

Alternatives: 6

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

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

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]

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%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;z \leq -1.5 \cdot 10^{+17} \lor \neg \left(z \leq 21\right):\\ \;\;\;\;t\_1 - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(\log t - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (or (<= z -1.5e+17) (not (<= z 21.0)))
     (- t_1 (+ y z))
     (+ t_1 (- (log t) y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if ((z <= -1.5e+17) || !(z <= 21.0)) {
		tmp = t_1 - (y + z);
	} else {
		tmp = t_1 + (log(t) - y);
	}
	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 ((z <= (-1.5d+17)) .or. (.not. (z <= 21.0d0))) then
        tmp = t_1 - (y + z)
    else
        tmp = t_1 + (log(t) - y)
    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 ((z <= -1.5e+17) || !(z <= 21.0)) {
		tmp = t_1 - (y + z);
	} else {
		tmp = t_1 + (Math.log(t) - y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if (z <= -1.5e+17) or not (z <= 21.0):
		tmp = t_1 - (y + z)
	else:
		tmp = t_1 + (math.log(t) - y)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if ((z <= -1.5e+17) || !(z <= 21.0))
		tmp = Float64(t_1 - Float64(y + z));
	else
		tmp = Float64(t_1 + Float64(log(t) - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if ((z <= -1.5e+17) || ~((z <= 21.0)))
		tmp = t_1 - (y + z);
	else
		tmp = t_1 + (log(t) - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[z, -1.5e+17], N[Not[LessEqual[z, 21.0]], $MachinePrecision]], N[(t$95$1 - N[(y + z), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5e17 or 21 < z

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. associate--l- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 99.4%

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

   if -1.5e17 < z < 21

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

      2. associate--l- 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+17} \lor \neg \left(z \leq 21\right):\\ \;\;\;\;x \cdot \log y - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y + \left(\log t - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{+108}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 1.6e+108) (* x (log y)) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.6e+108) {
		tmp = x * log(y);
	} else {
		tmp = -y;
	}
	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 <= 1.6d+108) then
        tmp = x * log(y)
    else
        tmp = -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.6e+108) {
		tmp = x * Math.log(y);
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1.6e+108:
		tmp = x * math.log(y)
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.6e+108)
		tmp = Float64(x * log(y));
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.6e+108)
		tmp = x * log(y);
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.6e+108], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 1.6e108

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. associate--l- 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 46.8%

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

      1. add-log-exp 10.7%

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

      2. *-commutative 10.7%

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

      3. exp-to-pow 10.7%

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

   7. Applied egg-rr 10.7%

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

   8. Taylor expanded in y around 0 4.7%

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

      1. log-pow 39.2%

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

      2. *-commutative 39.2%

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

   10. Applied egg-rr 39.2%

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

   if 1.6e108 < y

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. associate--l- 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 86.3%

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

      1. add-log-exp 50.2%

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

      2. *-commutative 50.2%

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

      3. exp-to-pow 50.2%

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

   7. Applied egg-rr 50.2%

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

   8. Taylor expanded in y around inf 76.4%

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

      1. neg-mul-1 76.4%

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

   10. Simplified 76.4%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{+108}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l- 99.9%

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

   2. associate--l- 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in z around inf 87.1%

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

6. Final simplification 87.1%

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

Alternative 5: 58.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \log y - y \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l- 99.9%

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

   2. associate--l- 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in y around inf 59.3%

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

6. Final simplification 59.3%

   \[\leadsto x \cdot \log y - y \]
7. Add Preprocessing

Alternative 6: 29.6% accurate, 104.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l- 99.9%

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

   2. associate--l- 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in y around inf 59.3%

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

   1. add-log-exp 23.2%

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

   2. *-commutative 23.2%

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

   3. exp-to-pow 23.2%

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

7. Applied egg-rr 23.2%

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

8. Taylor expanded in y around inf 30.7%

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

   1. neg-mul-1 30.7%

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

10. Simplified 30.7%

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

11. Final simplification 30.7%

    \[\leadsto -y \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024079 
(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)))