Average Error: 20.1 → 0.3
Time: 12.7s
Precision: binary64
\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
↓
\[\begin{array}{l} t_0 := \mathsf{hypot}\left(x, \sqrt{x}\right)\\ \frac{\frac{1}{t_0}}{t_0 \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}\right)} \end{array} \]
(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (hypot x (sqrt x))))
   (/ (/ 1.0 t_0) (* t_0 (+ (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ 1.0 x))))))))
double code(double x) {
	return (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
}
↓
double code(double x) {
	double t_0 = hypot(x, sqrt(x));
	return (1.0 / t_0) / (t_0 * ((1.0 / sqrt(x)) + (1.0 / sqrt((1.0 + x)))));
}

public static double code(double x) {
	return (1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((x + 1.0)));
}
↓
public static double code(double x) {
	double t_0 = Math.hypot(x, Math.sqrt(x));
	return (1.0 / t_0) / (t_0 * ((1.0 / Math.sqrt(x)) + (1.0 / Math.sqrt((1.0 + x)))));
}

def code(x):
	return (1.0 / math.sqrt(x)) - (1.0 / math.sqrt((x + 1.0)))
↓
def code(x):
	t_0 = math.hypot(x, math.sqrt(x))
	return (1.0 / t_0) / (t_0 * ((1.0 / math.sqrt(x)) + (1.0 / math.sqrt((1.0 + x)))))

function code(x)
	return Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0))))
end
↓
function code(x)
	t_0 = hypot(x, sqrt(x))
	return Float64(Float64(1.0 / t_0) / Float64(t_0 * Float64(Float64(1.0 / sqrt(x)) + Float64(1.0 / sqrt(Float64(1.0 + x))))))
end

function tmp = code(x)
	tmp = (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
end
↓
function tmp = code(x)
	t_0 = hypot(x, sqrt(x));
	tmp = (1.0 / t_0) / (t_0 * ((1.0 / sqrt(x)) + (1.0 / sqrt((1.0 + 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_] := Block[{t$95$0 = N[Sqrt[x ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] / N[(t$95$0 * N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}
↓
\begin{array}{l}
t_0 := \mathsf{hypot}\left(x, \sqrt{x}\right)\\
\frac{\frac{1}{t_0}}{t_0 \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}\right)}
\end{array}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.1
Target0.7
Herbie0.3
\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \]

Derivation

  1. Initial program 20.1

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

  2. Applied flip--_binary6420.1

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

  3. Simplified20.1

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

  4. Simplified20.1

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

  5. Applied frac-sub_binary6419.5

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

  6. Simplified5.6

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

  7. Simplified5.5

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

  8. Applied *-un-lft-identity_binary645.5

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

  9. Applied add-sqr-sqrt_binary645.6

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

  10. Applied add-cube-cbrt_binary645.6

    \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\sqrt{\mathsf{fma}\left(x, x, x\right)} \cdot \sqrt{\mathsf{fma}\left(x, x, x\right)}}}{1 \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}\right)} \]

  11. Applied times-frac_binary645.5

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{\mathsf{fma}\left(x, x, x\right)}} \cdot \frac{\sqrt[3]{1}}{\sqrt{\mathsf{fma}\left(x, x, x\right)}}}}{1 \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}\right)} \]

  12. Applied times-frac_binary645.5

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{\mathsf{fma}\left(x, x, x\right)}}}{1} \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt{\mathsf{fma}\left(x, x, x\right)}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}}} \]

  13. Simplified5.5

    \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right)}} \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt{\mathsf{fma}\left(x, x, x\right)}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}} \]

  14. Simplified0.4

    \[\leadsto \frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right)} \cdot \color{blue}{\frac{\frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right)}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}}} \]

  15. Applied frac-times_binary640.3

    \[\leadsto \color{blue}{\frac{1 \cdot \frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right)}}{\mathsf{hypot}\left(x, \sqrt{x}\right) \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}\right)}} \]

  16. Final simplification0.3

    \[\leadsto \frac{\frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right)}}{\mathsf{hypot}\left(x, \sqrt{x}\right) \cdot \left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}\right)} \]

Reproduce

herbie shell --seed 2022129 
(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)))))