Average Error: 19.5 → 0.2
Time: 4.6s
Precision: binary64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
↓
\[\frac{{\left(1 + x\right)}^{-0.5}}{\mathsf{fma}\left(\sqrt{1 + x}, \sqrt{x}, x\right)} \]
(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (/ (pow (+ 1.0 x) -0.5) (fma (sqrt (+ 1.0 x)) (sqrt x) x)))
double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
}
↓
double code(double x) {
	return pow((1.0 + x), -0.5) / fma(sqrt((1.0 + x)), sqrt(x), x);
}

function code(x)
	return Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0))))
end
↓
function code(x)
	return Float64((Float64(1.0 + x) ^ -0.5) / fma(sqrt(Float64(1.0 + x)), sqrt(x), x))
end

code[x_] := N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision] / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[x], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]

\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\frac{{\left(1 + x\right)}^{-0.5}}{\mathsf{fma}\left(\sqrt{1 + x}, \sqrt{x}, x\right)}

Error

Bits error versus x

Target

Original19.5
Target0.7
Herbie0.2
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \]

Derivation

  1. Initial program 19.5

    \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

  2. Applied egg-rr19.5

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}}}{\sqrt{1 + x}}} \]

  3. Applied egg-rr18.8

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

  4. Taylor expanded in x around 0 0.4

    \[\leadsto \frac{\frac{\frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{x}}}{\sqrt{1 + x}} \]

  5. Applied egg-rr0.2

    \[\leadsto \color{blue}{\frac{{\left(1 + x\right)}^{-0.5}}{\mathsf{fma}\left(\sqrt{1 + x}, \sqrt{x}, x\right)}} \]

  6. Final simplification0.2

    \[\leadsto \frac{{\left(1 + x\right)}^{-0.5}}{\mathsf{fma}\left(\sqrt{1 + x}, \sqrt{x}, x\right)} \]

Reproduce

herbie shell --seed 2022133 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))