Average Error: 18.2 → 0.1
Time: 4.6s
Precision: binary64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
↓
\[\log \left(\frac{1}{1 - x} \cdot \left(e \cdot \left(1 - y\right)\right)\right) \]
(FPCore (x y) :precision binary64 (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))
↓
(FPCore (x y) :precision binary64 (log (* (/ 1.0 (- 1.0 x)) (* E (- 1.0 y)))))
double code(double x, double y) {
	return 1.0 - log((1.0 - ((x - y) / (1.0 - y))));
}
↓
double code(double x, double y) {
	return log(((1.0 / (1.0 - x)) * (((double) M_E) * (1.0 - y))));
}

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) {
	return Math.log(((1.0 / (1.0 - x)) * (Math.E * (1.0 - y))));
}

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

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)
	return log(Float64(Float64(1.0 / Float64(1.0 - x)) * Float64(exp(1) * Float64(1.0 - y))))
end

function tmp = code(x, y)
	tmp = 1.0 - log((1.0 - ((x - y) / (1.0 - y))));
end
↓
function tmp = code(x, y)
	tmp = log(((1.0 / (1.0 - x)) * (2.71828182845904523536 * (1.0 - y))));
end

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_] := N[Log[N[(N[(1.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision] * N[(E * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\log \left(\frac{1}{1 - x} \cdot \left(e \cdot \left(1 - y\right)\right)\right)

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.2
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y < -81284752.61947241:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y < 3.0094271212461764 \cdot 10^{+25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array} \]

Derivation

  1. Initial program 18.2

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

  2. Simplified18.2

    \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]

  3. Applied egg-rr18.2

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

  4. Taylor expanded in y around 0 0.1

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

  5. Simplified0.1

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

  6. Applied egg-rr0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2022162 
(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))))))