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

Average Error: 17.9 → 0.1

Time: 19.6s

Precision: binary64

Cost: 20420

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -650000:\\ \;\;\;\;1 - \log \left(\left(\frac{x}{{y}^{2}} + \frac{-1 + x}{y}\right) - \frac{1}{{y}^{2}}\right)\\ \mathbf{elif}\;y \leq 0.135:\\ \;\;\;\;\frac{\left(2 + \left(-\log \left(\frac{x - y}{\left(y + y\right) + -2} + \frac{2 + \frac{x - y}{y + -1}}{2}\right)\right)\right) + \left(-\log \left(1 - \frac{x - y}{1 - y}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{-1 + y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -650000.0)
   (- 1.0 (log (- (+ (/ x (pow y 2.0)) (/ (+ -1.0 x) y)) (/ 1.0 (pow y 2.0)))))
   (if (<= y 0.135)
     (/
      (+
       (+
        2.0
        (-
         (log
          (+
           (/ (- x y) (+ (+ y y) -2.0))
           (/ (+ 2.0 (/ (- x y) (+ y -1.0))) 2.0)))))
       (- (log (- 1.0 (/ (- x y) (- 1.0 y))))))
      2.0)
     (- 1.0 (log (/ x (+ -1.0 y)))))))

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 tmp;
	if (y <= -650000.0) {
		tmp = 1.0 - log((((x / pow(y, 2.0)) + ((-1.0 + x) / y)) - (1.0 / pow(y, 2.0))));
	} else if (y <= 0.135) {
		tmp = ((2.0 + -log((((x - y) / ((y + y) + -2.0)) + ((2.0 + ((x - y) / (y + -1.0))) / 2.0)))) + -log((1.0 - ((x - y) / (1.0 - y))))) / 2.0;
	} else {
		tmp = 1.0 - log((x / (-1.0 + y)));
	}
	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) :: tmp
    if (y <= (-650000.0d0)) then
        tmp = 1.0d0 - log((((x / (y ** 2.0d0)) + (((-1.0d0) + x) / y)) - (1.0d0 / (y ** 2.0d0))))
    else if (y <= 0.135d0) then
        tmp = ((2.0d0 + -log((((x - y) / ((y + y) + (-2.0d0))) + ((2.0d0 + ((x - y) / (y + (-1.0d0)))) / 2.0d0)))) + -log((1.0d0 - ((x - y) / (1.0d0 - y))))) / 2.0d0
    else
        tmp = 1.0d0 - log((x / ((-1.0d0) + y)))
    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 tmp;
	if (y <= -650000.0) {
		tmp = 1.0 - Math.log((((x / Math.pow(y, 2.0)) + ((-1.0 + x) / y)) - (1.0 / Math.pow(y, 2.0))));
	} else if (y <= 0.135) {
		tmp = ((2.0 + -Math.log((((x - y) / ((y + y) + -2.0)) + ((2.0 + ((x - y) / (y + -1.0))) / 2.0)))) + -Math.log((1.0 - ((x - y) / (1.0 - y))))) / 2.0;
	} else {
		tmp = 1.0 - Math.log((x / (-1.0 + y)));
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	tmp = 0
	if y <= -650000.0:
		tmp = 1.0 - math.log((((x / math.pow(y, 2.0)) + ((-1.0 + x) / y)) - (1.0 / math.pow(y, 2.0))))
	elif y <= 0.135:
		tmp = ((2.0 + -math.log((((x - y) / ((y + y) + -2.0)) + ((2.0 + ((x - y) / (y + -1.0))) / 2.0)))) + -math.log((1.0 - ((x - y) / (1.0 - y))))) / 2.0
	else:
		tmp = 1.0 - math.log((x / (-1.0 + y)))
	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)
	tmp = 0.0
	if (y <= -650000.0)
		tmp = Float64(1.0 - log(Float64(Float64(Float64(x / (y ^ 2.0)) + Float64(Float64(-1.0 + x) / y)) - Float64(1.0 / (y ^ 2.0)))));
	elseif (y <= 0.135)
		tmp = Float64(Float64(Float64(2.0 + Float64(-log(Float64(Float64(Float64(x - y) / Float64(Float64(y + y) + -2.0)) + Float64(Float64(2.0 + Float64(Float64(x - y) / Float64(y + -1.0))) / 2.0))))) + Float64(-log(Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y)))))) / 2.0);
	else
		tmp = Float64(1.0 - log(Float64(x / Float64(-1.0 + y))));
	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)
	tmp = 0.0;
	if (y <= -650000.0)
		tmp = 1.0 - log((((x / (y ^ 2.0)) + ((-1.0 + x) / y)) - (1.0 / (y ^ 2.0))));
	elseif (y <= 0.135)
		tmp = ((2.0 + -log((((x - y) / ((y + y) + -2.0)) + ((2.0 + ((x - y) / (y + -1.0))) / 2.0)))) + -log((1.0 - ((x - y) / (1.0 - y))))) / 2.0;
	else
		tmp = 1.0 - log((x / (-1.0 + y)));
	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_] := If[LessEqual[y, -650000.0], N[(1.0 - N[Log[N[(N[(N[(x / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.135], N[(N[(N[(2.0 + (-N[Log[N[(N[(N[(x - y), $MachinePrecision] / N[(N[(y + y), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] + (-N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 - N[Log[N[(x / N[(-1.0 + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Target

Original	17.9
Target	0.1
Herbie	0.1

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

Derivation

1. Split input into 3 regimes

2. if y < -6.5e5

   1. Initial program 51.3

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

   2. Taylor expanded in y around -inf 0.0

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

   3. Simplified 0.0

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

      [Start] 0.0	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} + -1 \cdot \frac{1 - x}{y}\right) - \frac{1}{{y}^{2}}\right) \]
      rational_best-simplify-62 [=>] 0.0	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} + \color{blue}{\frac{\left(1 - x\right) \cdot -1}{y}}\right) - \frac{1}{{y}^{2}}\right) \]
      metadata-eval [<=] 0.0	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} + \frac{\left(1 - x\right) \cdot \color{blue}{\left(-1\right)}}{y}\right) - \frac{1}{{y}^{2}}\right) \]
      rational_best-simplify-68 [<=] 0.0	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} + \frac{\color{blue}{1 \cdot \left(x - 1\right)}}{y}\right) - \frac{1}{{y}^{2}}\right) \]
      rational_best-simplify-1 [=>] 0.0	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} + \frac{\color{blue}{\left(x - 1\right) \cdot 1}}{y}\right) - \frac{1}{{y}^{2}}\right) \]
      rational_best-simplify-7 [=>] 0.0	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} + \frac{\color{blue}{x - 1}}{y}\right) - \frac{1}{{y}^{2}}\right) \]
      rational_best-simplify-18 [=>] 0.0	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} + \frac{\color{blue}{x + -1}}{y}\right) - \frac{1}{{y}^{2}}\right) \]
      rational_best-simplify-3 [=>] 0.0	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} + \frac{\color{blue}{-1 + x}}{y}\right) - \frac{1}{{y}^{2}}\right) \]

   if -6.5e5 < y < 0.13500000000000001

   1. Initial program 0.0

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

   2. Applied egg-rr 0.1

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

   3. Simplified 0.1

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

      [Start] 0.1	\[ \frac{-\log \left(1 - \frac{x - y}{1 - y}\right)}{2} + \frac{2 + \left(-\log \left(1 - \frac{x - y}{1 - y}\right)\right)}{2} \]
      rational_best-simplify-3 [<=] 0.1	\[ \color{blue}{\frac{2 + \left(-\log \left(1 - \frac{x - y}{1 - y}\right)\right)}{2} + \frac{-\log \left(1 - \frac{x - y}{1 - y}\right)}{2}} \]
      rational_best-simplify-76 [=>] 0.1	\[ \color{blue}{\frac{\left(2 + \left(-\log \left(1 - \frac{x - y}{1 - y}\right)\right)\right) + \left(-\log \left(1 - \frac{x - y}{1 - y}\right)\right)}{2}} \]

   4. Applied egg-rr 0.1

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

   if 0.13500000000000001 < y

   1. Initial program 29.5

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

   2. Taylor expanded in x around inf 0.9

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

   3. Simplified 0.9

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

      [Start] 0.9	\[ 1 - \log \left(-1 \cdot \frac{x}{1 - y}\right) \]
      rational_best-simplify-62 [=>] 0.9	\[ 1 - \log \color{blue}{\left(\frac{x \cdot -1}{1 - y}\right)} \]
      rational_best-simplify-10 [=>] 0.9	\[ 1 - \log \left(\frac{\color{blue}{-x}}{1 - y}\right) \]

   4. Taylor expanded in x around 0 0.9

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

   5. Simplified 0.9

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

      [Start] 0.9	\[ 1 - \log \left(-1 \cdot \frac{x}{1 - y}\right) \]
      rational_best-simplify-1 [<=] 0.9	\[ 1 - \log \color{blue}{\left(\frac{x}{1 - y} \cdot -1\right)} \]
      rational_best-simplify-10 [=>] 0.9	\[ 1 - \log \color{blue}{\left(-\frac{x}{1 - y}\right)} \]
      rational_best-simplify-13 [=>] 0.9	\[ 1 - \log \color{blue}{\left(\frac{\frac{x}{1 - y}}{-1}\right)} \]
      rational_best-simplify-59 [=>] 0.9	\[ 1 - \log \color{blue}{\left(\frac{x}{\left(1 - y\right) \cdot -1}\right)} \]
      rational_best-simplify-11 [<=] 0.9	\[ 1 - \log \left(\frac{x}{\color{blue}{-\left(1 - y\right)}}\right) \]
      rational_best-simplify-14 [=>] 0.9	\[ 1 - \log \left(\frac{x}{\color{blue}{0 - \left(1 - y\right)}}\right) \]
      rational_best-simplify-55 [=>] 0.9	\[ 1 - \log \left(\frac{x}{\color{blue}{y - \left(1 - 0\right)}}\right) \]
      metadata-eval [=>] 0.9	\[ 1 - \log \left(\frac{x}{y - \color{blue}{1}}\right) \]
      rational_best-simplify-19 [<=] 0.9	\[ 1 - \log \left(\frac{x}{\color{blue}{y + -1}}\right) \]
      rational_best-simplify-3 [=>] 0.9	\[ 1 - \log \left(\frac{x}{\color{blue}{-1 + y}}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -650000:\\ \;\;\;\;1 - \log \left(\left(\frac{x}{{y}^{2}} + \frac{-1 + x}{y}\right) - \frac{1}{{y}^{2}}\right)\\ \mathbf{elif}\;y \leq 0.135:\\ \;\;\;\;\frac{\left(2 + \left(-\log \left(\frac{x - y}{\left(y + y\right) + -2} + \frac{2 + \frac{x - y}{y + -1}}{2}\right)\right)\right) + \left(-\log \left(1 - \frac{x - y}{1 - y}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{-1 + y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	15432

\[\begin{array}{l} \mathbf{if}\;y \leq -1400000000:\\ \;\;\;\;1 - \log \left(\frac{-1 + x}{y}\right)\\ \mathbf{elif}\;y \leq 0.135:\\ \;\;\;\;\frac{\left(2 + \left(-\log \left(\frac{x - y}{\left(y + y\right) + -2} + \frac{2 + \frac{x - y}{y + -1}}{2}\right)\right)\right) + \left(-\log \left(1 - \frac{x - y}{1 - y}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{-1 + y}\right)\\ \end{array} \]

Alternative 2
Error	0.2
Cost	7752

\[\begin{array}{l} \mathbf{if}\;y \leq -6200000000:\\ \;\;\;\;1 - \log \left(\frac{-1 + x}{y}\right)\\ \mathbf{elif}\;y \leq 0.135:\\ \;\;\;\;\frac{-2 - \left(2 \cdot \log \left(1 + \frac{y - x}{1 - y}\right) - 4\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{-1 + y}\right)\\ \end{array} \]

Alternative 3
Error	0.2
Cost	7368

\[\begin{array}{l} \mathbf{if}\;y \leq -1450000000:\\ \;\;\;\;1 - \log \left(\frac{-1 + x}{y}\right)\\ \mathbf{elif}\;y \leq 0.135:\\ \;\;\;\;1 - \log \left(1 - \frac{x - y}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{-1 + y}\right)\\ \end{array} \]

Alternative 4
Error	6.7
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -106000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-5}:\\ \;\;\;\;1 - \log \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{-1 + y}\right)\\ \end{array} \]

Alternative 5
Error	0.9
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1 - \log \left(\frac{-1 + x}{y}\right)\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-5}:\\ \;\;\;\;1 - \log \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{-1 + y}\right)\\ \end{array} \]

Alternative 6
Error	6.9
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -106000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \log \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 7
Error	13.2
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -106000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - x\right)\\ \end{array} \]

Alternative 8
Error	23.8
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0009:\\ \;\;\;\;1 - \log \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \]

Alternative 9
Error	23.5
Cost	6720

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

Alternative 10
Error	36.1
Cost	192

\[x + 1 \]

Alternative 11
Error	36.3
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023101 
(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))))))