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

Average Error: 19.1 → 0.0

Time: 1.4min

Precision: binary64

Cost: 40900

Math

\[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]

↓

\[\begin{array}{l} t_0 := -1 \cdot \left(\frac{1 - x}{y} + \frac{1 + \left(-x\right)}{{y}^{3}}\right)\\ t_1 := \frac{x}{{y}^{2}}\\ t_2 := \frac{1}{{y}^{2}}\\ \mathbf{if}\;y \leq -1780:\\ \;\;\;\;1 - \log \left(\left(t_1 + \left(t_0 + \frac{x}{{y}^{4}}\right)\right) - \left(t_2 + \frac{1}{{y}^{4}}\right)\right)\\ \mathbf{elif}\;y \leq 1000:\\ \;\;\;\;1 - \log \left(1 - \left(\left(1 - \frac{x - y}{y + -1}\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(t_1 + t_0\right) - t_2\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* -1.0 (+ (/ (- 1.0 x) y) (/ (+ 1.0 (- x)) (pow y 3.0)))))
        (t_1 (/ x (pow y 2.0)))
        (t_2 (/ 1.0 (pow y 2.0))))
   (if (<= y -1780.0)
     (-
      1.0
      (log (- (+ t_1 (+ t_0 (/ x (pow y 4.0)))) (+ t_2 (/ 1.0 (pow y 4.0))))))
     (if (<= y 1000.0)
       (- 1.0 (log (- 1.0 (+ (- 1.0 (/ (- x y) (+ y -1.0))) -1.0))))
       (- 1.0 (log (- (+ t_1 t_0) t_2)))))))

C

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

↓

double code(double x, double y) {
	double t_0 = -1.0 * (((1.0 - x) / y) + ((1.0 + -x) / pow(y, 3.0)));
	double t_1 = x / pow(y, 2.0);
	double t_2 = 1.0 / pow(y, 2.0);
	double tmp;
	if (y <= -1780.0) {
		tmp = 1.0 - log(((t_1 + (t_0 + (x / pow(y, 4.0)))) - (t_2 + (1.0 / pow(y, 4.0)))));
	} else if (y <= 1000.0) {
		tmp = 1.0 - log((1.0 - ((1.0 - ((x - y) / (y + -1.0))) + -1.0)));
	} else {
		tmp = 1.0 - log(((t_1 + t_0) - t_2));
	}
	return tmp;
}

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

↓

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) :: t_2
    real(8) :: tmp
    t_0 = (-1.0d0) * (((1.0d0 - x) / y) + ((1.0d0 + -x) / (y ** 3.0d0)))
    t_1 = x / (y ** 2.0d0)
    t_2 = 1.0d0 / (y ** 2.0d0)
    if (y <= (-1780.0d0)) then
        tmp = 1.0d0 - log(((t_1 + (t_0 + (x / (y ** 4.0d0)))) - (t_2 + (1.0d0 / (y ** 4.0d0)))))
    else if (y <= 1000.0d0) then
        tmp = 1.0d0 - log((1.0d0 - ((1.0d0 - ((x - y) / (y + (-1.0d0)))) + (-1.0d0))))
    else
        tmp = 1.0d0 - log(((t_1 + t_0) - t_2))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y) {
	double t_0 = -1.0 * (((1.0 - x) / y) + ((1.0 + -x) / Math.pow(y, 3.0)));
	double t_1 = x / Math.pow(y, 2.0);
	double t_2 = 1.0 / Math.pow(y, 2.0);
	double tmp;
	if (y <= -1780.0) {
		tmp = 1.0 - Math.log(((t_1 + (t_0 + (x / Math.pow(y, 4.0)))) - (t_2 + (1.0 / Math.pow(y, 4.0)))));
	} else if (y <= 1000.0) {
		tmp = 1.0 - Math.log((1.0 - ((1.0 - ((x - y) / (y + -1.0))) + -1.0)));
	} else {
		tmp = 1.0 - Math.log(((t_1 + t_0) - t_2));
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	t_0 = -1.0 * (((1.0 - x) / y) + ((1.0 + -x) / math.pow(y, 3.0)))
	t_1 = x / math.pow(y, 2.0)
	t_2 = 1.0 / math.pow(y, 2.0)
	tmp = 0
	if y <= -1780.0:
		tmp = 1.0 - math.log(((t_1 + (t_0 + (x / math.pow(y, 4.0)))) - (t_2 + (1.0 / math.pow(y, 4.0)))))
	elif y <= 1000.0:
		tmp = 1.0 - math.log((1.0 - ((1.0 - ((x - y) / (y + -1.0))) + -1.0)))
	else:
		tmp = 1.0 - math.log(((t_1 + t_0) - t_2))
	return tmp

Julia

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

↓

function code(x, y)
	t_0 = Float64(-1.0 * Float64(Float64(Float64(1.0 - x) / y) + Float64(Float64(1.0 + Float64(-x)) / (y ^ 3.0))))
	t_1 = Float64(x / (y ^ 2.0))
	t_2 = Float64(1.0 / (y ^ 2.0))
	tmp = 0.0
	if (y <= -1780.0)
		tmp = Float64(1.0 - log(Float64(Float64(t_1 + Float64(t_0 + Float64(x / (y ^ 4.0)))) - Float64(t_2 + Float64(1.0 / (y ^ 4.0))))));
	elseif (y <= 1000.0)
		tmp = Float64(1.0 - log(Float64(1.0 - Float64(Float64(1.0 - Float64(Float64(x - y) / Float64(y + -1.0))) + -1.0))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(t_1 + t_0) - t_2)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y)
	t_0 = -1.0 * (((1.0 - x) / y) + ((1.0 + -x) / (y ^ 3.0)));
	t_1 = x / (y ^ 2.0);
	t_2 = 1.0 / (y ^ 2.0);
	tmp = 0.0;
	if (y <= -1780.0)
		tmp = 1.0 - log(((t_1 + (t_0 + (x / (y ^ 4.0)))) - (t_2 + (1.0 / (y ^ 4.0)))));
	elseif (y <= 1000.0)
		tmp = 1.0 - log((1.0 - ((1.0 - ((x - y) / (y + -1.0))) + -1.0)));
	else
		tmp = 1.0 - log(((t_1 + t_0) - t_2));
	end
	tmp_2 = tmp;
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]

↓

code[x_, y_] := Block[{t$95$0 = N[(-1.0 * N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] + N[(N[(1.0 + (-x)), $MachinePrecision] / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1780.0], N[(1.0 - N[Log[N[(N[(t$95$1 + N[(t$95$0 + N[(x / N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$2 + N[(1.0 / N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1000.0], N[(1.0 - N[Log[N[(1.0 - N[(N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(t$95$1 + t$95$0), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	19.1
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 3 regimes

2. if y < -1780

   1. Initial program 52.2

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]

   2. Taylor expanded in y around -inf 0.0

      \[\leadsto 1 - \log \color{blue}{\left(\left(\frac{x}{{y}^{4}} + \left(\frac{x}{{y}^{2}} + \left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{3}} + -1 \cdot \frac{1 - x}{y}\right)\right)\right) - \left(\frac{1}{{y}^{2}} + \frac{1}{{y}^{4}}\right)\right)} \]

   3. Simplified 0.0

      \[\leadsto 1 - \log \color{blue}{\left(\left(\frac{x}{{y}^{2}} + \left(-1 \cdot \left(\frac{1 - x}{y} + \frac{1 + \left(-x\right)}{{y}^{3}}\right) + \frac{x}{{y}^{4}}\right)\right) - \left(\frac{1}{{y}^{2}} + \frac{1}{{y}^{4}}\right)\right)} \]
      Proof

      [Start] 0.0	\[ 1 - \log \left(\left(\frac{x}{{y}^{4}} + \left(\frac{x}{{y}^{2}} + \left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{3}} + -1 \cdot \frac{1 - x}{y}\right)\right)\right) - \left(\frac{1}{{y}^{2}} + \frac{1}{{y}^{4}}\right)\right) \]

   if -1780 < y < 1e3

   1. Initial program 0.0

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]

   2. Applied egg-rr 0.0

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

   if 1e3 < y

   1. Initial program 31.1

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]

   2. Taylor expanded in y around -inf 0.0

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

   3. Simplified 0.0

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

      [Start] 0.0	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} + \left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{3}} + -1 \cdot \frac{1 - x}{y}\right)\right) - \frac{1}{{y}^{2}}\right) \]
      rational_best-simplify-3 [=>] 0.0	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} + \color{blue}{\left(-1 \cdot \frac{1 - x}{y} + -1 \cdot \frac{1 + -1 \cdot x}{{y}^{3}}\right)}\right) - \frac{1}{{y}^{2}}\right) \]
      rational_best-simplify-1 [=>] 0.0	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} + \left(\color{blue}{\frac{1 - x}{y} \cdot -1} + -1 \cdot \frac{1 + -1 \cdot x}{{y}^{3}}\right)\right) - \frac{1}{{y}^{2}}\right) \]
      rational_best-simplify-1 [=>] 0.0	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} + \left(\frac{1 - x}{y} \cdot -1 + \color{blue}{\frac{1 + -1 \cdot x}{{y}^{3}} \cdot -1}\right)\right) - \frac{1}{{y}^{2}}\right) \]
      rational_best-simplify-63 [=>] 0.0	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} + \color{blue}{-1 \cdot \left(\frac{1 - x}{y} + \frac{1 + -1 \cdot x}{{y}^{3}}\right)}\right) - \frac{1}{{y}^{2}}\right) \]
      rational_best-simplify-1 [=>] 0.0	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} + -1 \cdot \left(\frac{1 - x}{y} + \frac{1 + \color{blue}{x \cdot -1}}{{y}^{3}}\right)\right) - \frac{1}{{y}^{2}}\right) \]
      rational_best-simplify-10 [=>] 0.0	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} + -1 \cdot \left(\frac{1 - x}{y} + \frac{1 + \color{blue}{\left(-x\right)}}{{y}^{3}}\right)\right) - \frac{1}{{y}^{2}}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1780:\\ \;\;\;\;1 - \log \left(\left(\frac{x}{{y}^{2}} + \left(-1 \cdot \left(\frac{1 - x}{y} + \frac{1 + \left(-x\right)}{{y}^{3}}\right) + \frac{x}{{y}^{4}}\right)\right) - \left(\frac{1}{{y}^{2}} + \frac{1}{{y}^{4}}\right)\right)\\ \mathbf{elif}\;y \leq 1000:\\ \;\;\;\;1 - \log \left(1 - \left(\left(1 - \frac{x - y}{y + -1}\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{x}{{y}^{2}} + -1 \cdot \left(\frac{1 - x}{y} + \frac{1 + \left(-x\right)}{{y}^{3}}\right)\right) - \frac{1}{{y}^{2}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.0
Cost	27592

\[\begin{array}{l} t_0 := 1 - \log \left(\left(\frac{x}{{y}^{2}} + -1 \cdot \left(\frac{1 - x}{y} + \frac{1 + \left(-x\right)}{{y}^{3}}\right)\right) - \frac{1}{{y}^{2}}\right)\\ \mathbf{if}\;y \leq -12800:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 100000:\\ \;\;\;\;1 - \log \left(1 - \left(\left(1 - \frac{x - y}{y + -1}\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.2
Cost	14212

\[\begin{array}{l} \mathbf{if}\;y \leq -720000:\\ \;\;\;\;\frac{-\left(1 - x\right)}{y \cdot \left(1 - x\right)} - \left(-1 + \left(\log \left(1 - x\right) + \log \left(\frac{-1}{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 10000000:\\ \;\;\;\;1 - \log \left(1 - \frac{x - y}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \end{array} \]

Alternative 3
Error	7.7
Cost	7376

\[\begin{array}{l} t_0 := 1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{if}\;y \leq -8.2 \cdot 10^{+180}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -4.1 \cdot 10^{+156}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -22:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-8}:\\ \;\;\;\;1 - \log \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y - 1}\right)\\ \end{array} \]

Alternative 4
Error	0.2
Cost	7368

\[\begin{array}{l} t_0 := 1 - \log \left(\frac{x - 1}{y}\right)\\ \mathbf{if}\;y \leq -18000000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10000000:\\ \;\;\;\;1 - \log \left(1 - \frac{x - y}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	7.7
Cost	7248

\[\begin{array}{l} t_0 := 1 - \log \left(\frac{-1}{y}\right)\\ t_1 := 1 - \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;y \leq -8.2 \cdot 10^{+180}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{+156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -20.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \log \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	1.3
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-8}:\\ \;\;\;\;1 - \log \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y - 1}\right)\\ \end{array} \]

Alternative 7
Error	13.6
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -22:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - x\right)\\ \end{array} \]

Alternative 8
Error	24.9
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq -0.85:\\ \;\;\;\;1 - \log \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \]

Alternative 9
Error	24.6
Cost	6720

\[1 - \log \left(1 - x\right) \]

Alternative 10
Error	37.0
Cost	192

\[x - -1 \]

Alternative 11
Error	37.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023099 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< y -81284752.61947241) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1.0) (- 1.0 (/ (- x y) (- 1.0 y))))) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y)))))))

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