Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B

Percentage Accurate: 71.6% → 74.7%

Time: 2.1s

Alternatives: 3

Speedup: 0.8×

Specification

\[\mathsf{TRUE}\left(\right)\]

Math

\[\begin{array}{l} \\ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))

C

double code(double x, double y) {
	return 1.0 - log((1.0 - ((x - y) / (1.0 - y))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - log((1.0d0 - ((x - y) / (1.0d0 - y))))
end function

Java

public static double code(double x, double y) {
	return 1.0 - Math.log((1.0 - ((x - y) / (1.0 - y))));
}

Python

def code(x, y):
	return 1.0 - math.log((1.0 - ((x - y) / (1.0 - y))))

Julia

function code(x, y)
	return Float64(1.0 - log(Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y)))))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - log((1.0 - ((x - y) / (1.0 - y))));
end

Wolfram

code[x_, y_] := N[(1.0 - N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 71.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))

C

double code(double x, double y) {
	return 1.0 - log((1.0 - ((x - y) / (1.0 - y))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - log((1.0d0 - ((x - y) / (1.0d0 - y))))
end function

Java

public static double code(double x, double y) {
	return 1.0 - Math.log((1.0 - ((x - y) / (1.0 - y))));
}

Python

def code(x, y):
	return 1.0 - math.log((1.0 - ((x - y) / (1.0 - y))))

Julia

function code(x, y)
	return Float64(1.0 - log(Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y)))))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - log((1.0 - ((x - y) / (1.0 - y))));
end

Wolfram

code[x_, y_] := N[(1.0 - N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 74.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ t_1 := 1 - t\_0\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1 - \log t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))) (t_1 (- 1.0 t_0)))
   (if (<= t_1 0.0) t_0 (- 1.0 (log t_1)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double t_1 = 1.0 - t_0;
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 - log(t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x - y) / (1.0d0 - y)
    t_1 = 1.0d0 - t_0
    if (t_1 <= 0.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0 - log(t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double t_1 = 1.0 - t_0;
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 - Math.log(t_1);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (1.0 - y)
	t_1 = 1.0 - t_0
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_0
	else:
		tmp = 1.0 - math.log(t_1)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	t_1 = Float64(1.0 - t_0)
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	else
		tmp = Float64(1.0 - log(t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (1.0 - y);
	t_1 = 1.0 - t_0;
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_0;
	else
		tmp = 1.0 - log(t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$0, N[(1.0 - N[Log[t$95$1], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))) < 0.0

   1. Initial program 3.1%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{3} \cdot \frac{x}{{\left(1 + \frac{y}{1 - y}\right)}^{3} \cdot {\left(1 - y\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{{\left(1 + \frac{y}{1 - y}\right)}^{2} \cdot {\left(1 - y\right)}^{2}}\right) + \frac{1}{\left(1 + \frac{y}{1 - y}\right) \cdot \left(1 - y\right)}\right)\right) - \log \left(1 + \frac{y}{1 - y}\right)} \]

   5. Applied rewrites 14.0%

      \[\leadsto \color{blue}{\frac{x - y}{1 - y}} \]

   if 0.0 < (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))

   1. Initial program 98.8%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 43.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;1 - t\_0 \leq 0.2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y)))) (if (<= (- 1.0 t_0) 0.2) t_0 (- 1.0 y))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if ((1.0 - t_0) <= 0.2) {
		tmp = t_0;
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / (1.0d0 - y)
    if ((1.0d0 - t_0) <= 0.2d0) then
        tmp = t_0
    else
        tmp = 1.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if ((1.0 - t_0) <= 0.2) {
		tmp = t_0;
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (1.0 - y)
	tmp = 0
	if (1.0 - t_0) <= 0.2:
		tmp = t_0
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	tmp = 0.0
	if (Float64(1.0 - t_0) <= 0.2)
		tmp = t_0;
	else
		tmp = Float64(1.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (1.0 - y);
	tmp = 0.0;
	if ((1.0 - t_0) <= 0.2)
		tmp = t_0;
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 - t$95$0), $MachinePrecision], 0.2], t$95$0, N[(1.0 - y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))) < 0.20000000000000001

   1. Initial program 10.4%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{3} \cdot \frac{x}{{\left(1 + \frac{y}{1 - y}\right)}^{3} \cdot {\left(1 - y\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{{\left(1 + \frac{y}{1 - y}\right)}^{2} \cdot {\left(1 - y\right)}^{2}}\right) + \frac{1}{\left(1 + \frac{y}{1 - y}\right) \cdot \left(1 - y\right)}\right)\right) - \log \left(1 + \frac{y}{1 - y}\right)} \]

   5. Applied rewrites 14.2%

      \[\leadsto \color{blue}{\frac{x - y}{1 - y}} \]

   if 0.20000000000000001 < (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{3} \cdot \frac{x}{{\left(1 + \frac{y}{1 - y}\right)}^{3} \cdot {\left(1 - y\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{{\left(1 + \frac{y}{1 - y}\right)}^{2} \cdot {\left(1 - y\right)}^{2}}\right) + \frac{1}{\left(1 + \frac{y}{1 - y}\right) \cdot \left(1 - y\right)}\right)\right) - \log \left(1 + \frac{y}{1 - y}\right)} \]

   5. Applied rewrites 5.0%

      \[\leadsto \color{blue}{\frac{x - y}{1 - y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \frac{y}{1 - y} + \color{blue}{\frac{x}{1 - y}} \]

   8. Applied rewrites 57.8%

      \[\leadsto 1 - \color{blue}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 40.5% accurate, 31.0× speedup

Math

\[\begin{array}{l} \\ 1 - y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 y))

C

double code(double x, double y) {
	return 1.0 - y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - y
end function

Java

public static double code(double x, double y) {
	return 1.0 - y;
}

Python

def code(x, y):
	return 1.0 - y

Julia

function code(x, y)
	return Float64(1.0 - y)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - y;
end

Wolfram

code[x_, y_] := N[(1.0 - y), $MachinePrecision]

Derivation

1. Initial program 72.7%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{3} \cdot \frac{x}{{\left(1 + \frac{y}{1 - y}\right)}^{3} \cdot {\left(1 - y\right)}^{3}} + \frac{1}{2} \cdot \frac{1}{{\left(1 + \frac{y}{1 - y}\right)}^{2} \cdot {\left(1 - y\right)}^{2}}\right) + \frac{1}{\left(1 + \frac{y}{1 - y}\right) \cdot \left(1 - y\right)}\right)\right) - \log \left(1 + \frac{y}{1 - y}\right)} \]

5. Applied rewrites 7.8%

   \[\leadsto \color{blue}{\frac{x - y}{1 - y}} \]

6. Taylor expanded in x around 0

   \[\leadsto -1 \cdot \frac{y}{1 - y} + \color{blue}{\frac{x}{1 - y}} \]

8. Applied rewrites 41.5%

   \[\leadsto 1 - \color{blue}{y} \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{if}\;y < -81284752.61947241:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 3.0094271212461764 \cdot 10^{+25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y)))))))
   (if (< y -81284752.61947241)
     t_0
     (if (< y 3.0094271212461764e+25)
       (log (/ (exp 1.0) (- 1.0 (/ (- x y) (- 1.0 y)))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log(((x / (y * y)) - ((1.0 / y) - (x / y))));
	double tmp;
	if (y < -81284752.61947241) {
		tmp = t_0;
	} else if (y < 3.0094271212461764e+25) {
		tmp = log((exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - log(((x / (y * y)) - ((1.0d0 / y) - (x / y))))
    if (y < (-81284752.61947241d0)) then
        tmp = t_0
    else if (y < 3.0094271212461764d+25) then
        tmp = log((exp(1.0d0) / (1.0d0 - ((x - y) / (1.0d0 - y)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.log(((x / (y * y)) - ((1.0 / y) - (x / y))));
	double tmp;
	if (y < -81284752.61947241) {
		tmp = t_0;
	} else if (y < 3.0094271212461764e+25) {
		tmp = Math.log((Math.exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log(((x / (y * y)) - ((1.0 / y) - (x / y))))
	tmp = 0
	if y < -81284752.61947241:
		tmp = t_0
	elif y < 3.0094271212461764e+25:
		tmp = math.log((math.exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - log(Float64(Float64(x / Float64(y * y)) - Float64(Float64(1.0 / y) - Float64(x / y)))))
	tmp = 0.0
	if (y < -81284752.61947241)
		tmp = t_0;
	elseif (y < 3.0094271212461764e+25)
		tmp = log(Float64(exp(1.0) / Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - log(((x / (y * y)) - ((1.0 / y) - (x / y))));
	tmp = 0.0;
	if (y < -81284752.61947241)
		tmp = t_0;
	elseif (y < 3.0094271212461764e+25)
		tmp = log((exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / y), $MachinePrecision] - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Less[y, -81284752.61947241], t$95$0, If[Less[y, 3.0094271212461764e+25], N[Log[N[(N[Exp[1.0], $MachinePrecision] / N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024321 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64
  :pre (TRUE)

  :alt
  (! :herbie-platform default (if (< y -8128475261947241/100000000) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))) (if (< y 30094271212461764000000000) (log (/ (exp 1) (- 1 (/ (- x y) (- 1 y))))) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))))))

  (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))