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

Percentage Accurate: 99.9% → 99.9%

Time: 8.4s

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. Final simplification 99.9%

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

Alternative 2: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+132} \lor \neg \left(x \leq 195000000000\right):\\ \;\;\;\;\log t + \left(x \cdot \log y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.8e+132) (not (<= x 195000000000.0)))
   (+ (log t) (- (* x (log y)) z))
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.8e+132) || !(x <= 195000000000.0)) {
		tmp = log(t) + ((x * log(y)) - z);
	} else {
		tmp = log(t) - (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 ((x <= (-2.8d+132)) .or. (.not. (x <= 195000000000.0d0))) then
        tmp = log(t) + ((x * log(y)) - z)
    else
        tmp = log(t) - (y + z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.8e+132) || !(x <= 195000000000.0)) {
		tmp = Math.log(t) + ((x * Math.log(y)) - z);
	} else {
		tmp = Math.log(t) - (y + z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.8e+132) or not (x <= 195000000000.0):
		tmp = math.log(t) + ((x * math.log(y)) - z)
	else:
		tmp = math.log(t) - (y + z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.8e+132) || !(x <= 195000000000.0))
		tmp = Float64(log(t) + Float64(Float64(x * log(y)) - z));
	else
		tmp = Float64(log(t) - Float64(y + z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2.8e+132) || ~((x <= 195000000000.0)))
		tmp = log(t) + ((x * log(y)) - z);
	else
		tmp = log(t) - (y + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.8e+132], N[Not[LessEqual[x, 195000000000.0]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] + N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7999999999999999e132 or 1.95e11 < x

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 40.6%

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

      2. pow2 40.6%

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

   3. Applied egg-rr 40.6%

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

   4. Taylor expanded in y around 0 44.5%

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

      1. unpow2 44.5%

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

      2. add-sqr-sqrt 90.1%

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

      3. *-commutative 90.1%

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

   6. Applied egg-rr 90.1%

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

   if -2.7999999999999999e132 < x < 1.95e11

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 94.1%

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

      1. neg-mul-1 94.1%

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

   4. Simplified 94.1%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+132} \lor \neg \left(x \leq 195000000000\right):\\ \;\;\;\;\log t + \left(x \cdot \log y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 3: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.3 \cdot 10^{+132} \lor \neg \left(x \leq 6.8 \cdot 10^{+52}\right):\\ \;\;\;\;x \cdot \log y + \log t\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -5.3e+132) (not (<= x 6.8e+52)))
   (+ (* x (log y)) (log t))
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5.3e+132) || !(x <= 6.8e+52)) {
		tmp = (x * log(y)) + log(t);
	} else {
		tmp = log(t) - (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 ((x <= (-5.3d+132)) .or. (.not. (x <= 6.8d+52))) then
        tmp = (x * log(y)) + log(t)
    else
        tmp = log(t) - (y + z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5.3e+132) || !(x <= 6.8e+52)) {
		tmp = (x * Math.log(y)) + Math.log(t);
	} else {
		tmp = Math.log(t) - (y + z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -5.3e+132) or not (x <= 6.8e+52):
		tmp = (x * math.log(y)) + math.log(t)
	else:
		tmp = math.log(t) - (y + z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -5.3e+132) || !(x <= 6.8e+52))
		tmp = Float64(Float64(x * log(y)) + log(t));
	else
		tmp = Float64(log(t) - Float64(y + z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -5.3e+132) || ~((x <= 6.8e+52)))
		tmp = (x * log(y)) + log(t);
	else
		tmp = log(t) - (y + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -5.3e+132], N[Not[LessEqual[x, 6.8e+52]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[Log[t], $MachinePrecision]), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.3e132 or 6.8e52 < x

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 43.2%

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

      2. pow2 43.2%

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

   3. Applied egg-rr 43.2%

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

   4. Taylor expanded in y around 0 45.8%

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

      1. unpow2 45.8%

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

      2. add-sqr-sqrt 90.5%

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

      3. *-commutative 90.5%

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

   6. Applied egg-rr 90.5%

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

   7. Taylor expanded in z around 0 75.9%

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

   if -5.3e132 < x < 6.8e52

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 92.6%

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

      1. neg-mul-1 92.6%

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

   4. Simplified 92.6%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.3 \cdot 10^{+132} \lor \neg \left(x \leq 6.8 \cdot 10^{+52}\right):\\ \;\;\;\;x \cdot \log y + \log t\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 4: 70.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[Log[t], $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. Taylor expanded in x around 0 65.4%

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

   1. neg-mul-1 65.4%

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

4. Simplified 65.4%

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

5. Final simplification 65.4%

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

Alternative 5: 42.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[Log[t], $MachinePrecision] - z), $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. add-sqr-sqrt 23.3%

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

   2. pow2 23.3%

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

3. Applied egg-rr 23.3%

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

4. Taylor expanded in y around 0 32.0%

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

   1. unpow2 32.0%

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

   2. add-sqr-sqrt 71.7%

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

   3. *-commutative 71.7%

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

6. Applied egg-rr 71.7%

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

7. Taylor expanded in x around 0 37.4%

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

8. Final simplification 37.4%

   \[\leadsto \log t - z \]

Alternative 6: 30.4% accurate, 104.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. add-sqr-sqrt 23.3%

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

   2. pow2 23.3%

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

3. Applied egg-rr 23.3%

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

4. Taylor expanded in y around 0 32.0%

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

   1. unpow2 32.0%

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

   2. add-sqr-sqrt 71.7%

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

   3. *-commutative 71.7%

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

6. Applied egg-rr 71.7%

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

7. Taylor expanded in z around inf 30.5%

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

   1. neg-mul-1 30.5%

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

9. Simplified 30.5%

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

10. Final simplification 30.5%

    \[\leadsto -z \]

Reproduce

herbie shell --seed 2023268 
(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)))