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

Average Error: 0.34% → 0.42%

Time: 32.4s

Precision: binary64

Math

\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

↓

\[\left(1 - \frac{\frac{\sqrt[3]{9}}{x}}{\sqrt[3]{6561}}\right) - \frac{y}{\begin{array}{l} \mathbf{if}\;x \geq 0 \land 9 \geq 0:\\ \;\;\;\;\sqrt{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{NaN}\\ \end{array}} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))

↓

(FPCore (x y)
 :precision binary64
 (-
  (- 1.0 (/ (/ (cbrt 9.0) x) (cbrt 6561.0)))
  (/ y (if (and (>= x 0.0) (>= 9.0 0.0)) (sqrt (* x 9.0)) NAN))))

C

double code(double x, double y) {
	return (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * sqrt(x)));
}

↓

double code(double x, double y) {
	double tmp;
	if ((x >= 0.0) && (9.0 >= 0.0)) {
		tmp = sqrt((x * 9.0));
	} else {
		tmp = (double) NAN;
	}
	return (1.0 - ((cbrt(9.0) / x) / cbrt(6561.0))) - (y / tmp);
}

Java

public static double code(double x, double y) {
	return (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * Math.sqrt(x)));
}

↓

public static double code(double x, double y) {
	double tmp;
	if ((x >= 0.0) && (9.0 >= 0.0)) {
		tmp = Math.sqrt((x * 9.0));
	} else {
		tmp = Double.NaN;
	}
	return (1.0 - ((Math.cbrt(9.0) / x) / Math.cbrt(6561.0))) - (y / tmp);
}

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(1.0 / Float64(x * 9.0))) - Float64(y / Float64(3.0 * sqrt(x))))
end

↓

function code(x, y)
	tmp = 0.0
	if ((x >= 0.0) && (9.0 >= 0.0))
		tmp = sqrt(Float64(x * 9.0));
	else
		tmp = NaN;
	end
	return Float64(Float64(1.0 - Float64(Float64(cbrt(9.0) / x) / cbrt(6561.0))) - Float64(y / tmp))
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(N[(1.0 - N[(N[(N[Power[9.0, 1/3], $MachinePrecision] / x), $MachinePrecision] / N[Power[6561.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y / If[And[GreaterEqual[x, 0.0], GreaterEqual[9.0, 0.0]], N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision], Indeterminate]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.34%
Target	0.34%
Herbie	0.42%

\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

Derivation

1. Initial program 0.34

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]

2. Applied egg-rr 0.41

   \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;x \geq 0 \land 9 \geq 0:\\ \;\;\;\;\sqrt{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{NaN}\\ } \end{array}}} \]

3. Applied egg-rr 0.47

   \[\leadsto \left(1 - \color{blue}{\frac{\frac{{x}^{-1}}{\sqrt[3]{9}}}{\sqrt[3]{9 \cdot 9}}}\right) - \frac{y}{\begin{array}{l} \mathbf{if}\;x \geq 0 \land 9 \geq 0:\\ \;\;\;\;\sqrt{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{NaN}\\ \end{array}} \]

4. Simplified 0.47

   \[\leadsto \left(1 - \color{blue}{\frac{\frac{{x}^{-1}}{\sqrt[3]{9}}}{\sqrt[3]{81}}}\right) - \frac{y}{\begin{array}{l} \mathbf{if}\;x \geq 0 \land 9 \geq 0:\\ \;\;\;\;\sqrt{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{NaN}\\ \end{array}} \]
   Proof

5. Applied egg-rr 0.42

   \[\leadsto \left(1 - \color{blue}{\frac{\frac{\sqrt[3]{9}}{x}}{\sqrt[3]{81 \cdot 81}}}\right) - \frac{y}{\begin{array}{l} \mathbf{if}\;x \geq 0 \land 9 \geq 0:\\ \;\;\;\;\sqrt{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{NaN}\\ \end{array}} \]

6. Simplified 0.42

   \[\leadsto \left(1 - \color{blue}{\frac{\frac{\sqrt[3]{9}}{x}}{\sqrt[3]{6561}}}\right) - \frac{y}{\begin{array}{l} \mathbf{if}\;x \geq 0 \land 9 \geq 0:\\ \;\;\;\;\sqrt{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{NaN}\\ \end{array}} \]
   Proof

Reproduce

herbie shell --seed 2023136 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (- (- 1.0 (/ (/ 1.0 x) 9.0)) (/ y (* 3.0 (sqrt x))))

  (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))