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

Average Error: 18.0 → 0.1

Time: 19.1s

Precision: binary64

Cost: 27716

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t_0 \leq 0.5:\\ \;\;\;\;1 - \log \left(1 - t_0\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1 + x}{y} + \left(\frac{x}{{y}^{2}} + \left(\left(-\frac{1 - x}{{y}^{3}}\right) - \frac{1}{{y}^{2}}\right)\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= t_0 0.5)
     (- 1.0 (log (- 1.0 t_0)))
     (-
      1.0
      (log
       (+
        (/ (+ -1.0 x) y)
        (+
         (/ x (pow y 2.0))
         (- (- (/ (- 1.0 x) (pow y 3.0))) (/ 1.0 (pow y 2.0))))))))))

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 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= 0.5) {
		tmp = 1.0 - log((1.0 - t_0));
	} else {
		tmp = 1.0 - log((((-1.0 + x) / y) + ((x / pow(y, 2.0)) + (-((1.0 - x) / pow(y, 3.0)) - (1.0 / pow(y, 2.0))))));
	}
	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) :: tmp
    t_0 = (x - y) / (1.0d0 - y)
    if (t_0 <= 0.5d0) then
        tmp = 1.0d0 - log((1.0d0 - t_0))
    else
        tmp = 1.0d0 - log(((((-1.0d0) + x) / y) + ((x / (y ** 2.0d0)) + (-((1.0d0 - x) / (y ** 3.0d0)) - (1.0d0 / (y ** 2.0d0))))))
    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 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= 0.5) {
		tmp = 1.0 - Math.log((1.0 - t_0));
	} else {
		tmp = 1.0 - Math.log((((-1.0 + x) / y) + ((x / Math.pow(y, 2.0)) + (-((1.0 - x) / Math.pow(y, 3.0)) - (1.0 / Math.pow(y, 2.0))))));
	}
	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 = (x - y) / (1.0 - y)
	tmp = 0
	if t_0 <= 0.5:
		tmp = 1.0 - math.log((1.0 - t_0))
	else:
		tmp = 1.0 - math.log((((-1.0 + x) / y) + ((x / math.pow(y, 2.0)) + (-((1.0 - x) / math.pow(y, 3.0)) - (1.0 / math.pow(y, 2.0))))))
	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(Float64(x - y) / Float64(1.0 - y))
	tmp = 0.0
	if (t_0 <= 0.5)
		tmp = Float64(1.0 - log(Float64(1.0 - t_0)));
	else
		tmp = Float64(1.0 - log(Float64(Float64(Float64(-1.0 + x) / y) + Float64(Float64(x / (y ^ 2.0)) + Float64(Float64(-Float64(Float64(1.0 - x) / (y ^ 3.0))) - Float64(1.0 / (y ^ 2.0)))))));
	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 = (x - y) / (1.0 - y);
	tmp = 0.0;
	if (t_0 <= 0.5)
		tmp = 1.0 - log((1.0 - t_0));
	else
		tmp = 1.0 - log((((-1.0 + x) / y) + ((x / (y ^ 2.0)) + (-((1.0 - x) / (y ^ 3.0)) - (1.0 / (y ^ 2.0))))));
	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[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.5], N[(1.0 - N[Log[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(N[(-1.0 + x), $MachinePrecision] / y), $MachinePrecision] + N[(N[(x / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision] + N[((-N[(N[(1.0 - x), $MachinePrecision] / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]) - N[(1.0 / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Target

Original	18.0
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 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 0.5

   1. Initial program 0.0

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

   if 0.5 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 61.0

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

   2. Taylor expanded in y around -inf 0.4

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

   3. Simplified 0.4

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

      [Start] 0.4	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} + \left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{3}} + -1 \cdot \frac{1 - x}{y}\right)\right) - \frac{1}{{y}^{2}}\right) \]
      rational.json-simplify-48 [=>] 0.4	\[ 1 - \log \color{blue}{\left(\left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{3}} + -1 \cdot \frac{1 - x}{y}\right) + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right)} \]
      rational.json-simplify-1 [=>] 0.4	\[ 1 - \log \color{blue}{\left(\left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right) + \left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{3}} + -1 \cdot \frac{1 - x}{y}\right)\right)} \]
      rational.json-simplify-1 [=>] 0.4	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right) + \color{blue}{\left(-1 \cdot \frac{1 - x}{y} + -1 \cdot \frac{1 + -1 \cdot x}{{y}^{3}}\right)}\right) \]
      rational.json-simplify-41 [=>] 0.4	\[ 1 - \log \color{blue}{\left(-1 \cdot \frac{1 - x}{y} + \left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{3}} + \left(\frac{x}{{y}^{2}} - \frac{1}{{y}^{2}}\right)\right)\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.5:\\ \;\;\;\;1 - \log \left(1 - \frac{x - y}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1 + x}{y} + \left(\frac{x}{{y}^{2}} + \left(\left(-\frac{1 - x}{{y}^{3}}\right) - \frac{1}{{y}^{2}}\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	14468

\[\begin{array}{l} \mathbf{if}\;y \leq -520000:\\ \;\;\;\;1 - \left(\left(\log \left(1 - x\right) + \log \left(\frac{-1}{y}\right)\right) + \left(-\frac{\frac{x}{1 - x} - \frac{1}{1 - x}}{y}\right)\right)\\ \mathbf{elif}\;y \leq 2000000000:\\ \;\;\;\;1 - \log \left(1 - \frac{x - y}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 2
Error	0.2
Cost	7368

\[\begin{array}{l} \mathbf{if}\;y \leq -3600000000:\\ \;\;\;\;1 - \log \left(\frac{-1 + x}{y}\right)\\ \mathbf{elif}\;y \leq 2000000000:\\ \;\;\;\;1 - \log \left(1 - \frac{x - y}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 3
Error	7.2
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -28:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \log \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 4
Error	1.3
Cost	6984

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

Alternative 5
Error	13.0
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -175:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - x\right)\\ \end{array} \]

Alternative 6
Error	24.0
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq -27:\\ \;\;\;\;1 - \log \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	23.7
Cost	6720

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

Alternative 8
Error	36.0
Cost	256

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

Alternative 9
Error	37.7
Cost	192

\[1 - y \]

Reproduce

herbie shell --seed 2023063 
(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))))))