Average Error: 6.7 → 6.5
Time: 11.9s
Precision: binary64
Cost: 704
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\frac{1}{x}}{1 + z \cdot z}}{y}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\frac{1}{x}}{1 + z \cdot z}}{y}
(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))
↓
(FPCore (x y z) :precision binary64 (/ (/ (/ 1.0 x) (+ 1.0 (* z z))) y))
double code(double x, double y, double z) {
	return (1.0 / x) / (y * (1.0 + (z * z)));
}
↓
double code(double x, double y, double z) {
	return ((1.0 / x) / (1.0 + (z * z))) / y;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.7
Target6.0
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Alternatives

Alternative 1
Error7.2
Cost39616
\[\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{y} \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt[3]{x}}}{1 + z \cdot z}\]
Alternative 2
Error45.8
Cost27328
\[\left(\sqrt{\frac{1}{y}} \cdot \sqrt{\frac{\frac{1}{x}}{1 + z \cdot z}}\right) \cdot \left(\sqrt{\frac{1}{y}} \cdot \sqrt{\frac{\frac{1}{x}}{1 + z \cdot z}}\right)\]
Alternative 3
Error35.7
Cost26816
\[\frac{\frac{1}{x}}{\left(\sqrt{1 + z \cdot z} \cdot \sqrt{y}\right) \cdot \left(\sqrt{1 + z \cdot z} \cdot \sqrt{y}\right)}\]
Alternative 4
Error7.3
Cost21440
\[\sqrt[3]{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \cdot \left(\sqrt[3]{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \cdot \sqrt[3]{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}}\right)\]
Alternative 5
Error7.3
Cost20928
\[\frac{\frac{1}{x}}{\sqrt[3]{y \cdot \left(1 + z \cdot z\right)} \cdot \left(\sqrt[3]{y \cdot \left(1 + z \cdot z\right)} \cdot \sqrt[3]{y \cdot \left(1 + z \cdot z\right)}\right)}\]
Alternative 6
Error6.8
Cost20672
\[\frac{\frac{1}{x}}{\sqrt[3]{1 + z \cdot z} \cdot \left(y \cdot \left(\sqrt[3]{1 + z \cdot z} \cdot \sqrt[3]{1 + z \cdot z}\right)\right)}\]
Alternative 7
Error7.2
Cost20416
\[\frac{\sqrt[3]{\frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x}}}{y} \cdot \frac{\sqrt[3]{\frac{1}{x}}}{1 + z \cdot z}\]
Alternative 8
Error7.2
Cost20288
\[\frac{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{y} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{1 + z \cdot z}\]
Alternative 9
Error7.1
Cost20288
\[\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\frac{\frac{1}{x}}{1 + z \cdot z}}{\sqrt[3]{y}}\]
Alternative 10
Error7.3
Cost20160
\[\frac{\frac{1}{x}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\left(1 + z \cdot z\right) \cdot \sqrt[3]{y}\right)}\]
Alternative 11
Error6.8
Cost20160
\[\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{y}{\frac{\frac{\sqrt[3]{1}}{x}}{1 + z \cdot z}}}\]
Alternative 12
Error6.5
Cost20160
\[\frac{\frac{\sqrt[3]{1}}{x}}{1 + z \cdot z} \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{y}\]
Alternative 13
Error24.2
Cost14272
\[\sqrt{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \cdot \sqrt{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}}\]
Alternative 14
Error35.7
Cost14016
\[\frac{\frac{1}{x}}{\sqrt{y \cdot \left(1 + z \cdot z\right)} \cdot \sqrt{y \cdot \left(1 + z \cdot z\right)}}\]
Alternative 15
Error6.7
Cost13888
\[\frac{\frac{1}{x}}{\sqrt{1 + z \cdot z} \cdot \left(y \cdot \sqrt{1 + z \cdot z}\right)}\]
Alternative 16
Error35.6
Cost13760
\[\sqrt{\frac{1}{y}} \cdot \frac{\sqrt{\frac{1}{y}}}{x \cdot \left(1 + z \cdot z\right)}\]
Alternative 17
Error35.3
Cost13760
\[\frac{\sqrt{\frac{1}{x}}}{\frac{y}{\frac{\sqrt{\frac{1}{x}}}{1 + z \cdot z}}}\]
Alternative 18
Error35.4
Cost13760
\[\frac{\frac{1}{\sqrt{x}}}{y} \cdot \frac{\frac{1}{\sqrt{x}}}{1 + z \cdot z}\]
Alternative 19
Error35.3
Cost13760
\[\frac{\sqrt{\frac{1}{x}}}{1 + z \cdot z} \cdot \frac{\sqrt{\frac{1}{x}}}{y}\]
Alternative 20
Error26.3
Cost13696
\[\frac{1}{y} \cdot \sqrt[3]{\frac{1}{{\left(x \cdot \left(1 + z \cdot z\right)\right)}^{3}}}\]
Alternative 21
Error35.7
Cost13632
\[\frac{\frac{1}{x}}{\sqrt{y} \cdot \left(\left(1 + z \cdot z\right) \cdot \sqrt{y}\right)}\]
Alternative 22
Error35.5
Cost13632
\[\frac{\frac{1}{\sqrt{x}}}{\left(y \cdot \left(1 + z \cdot z\right)\right) \cdot \sqrt{x}}\]
Alternative 23
Error27.1
Cost13568
\[\sqrt[3]{\frac{1}{{\left(y \cdot \left(x \cdot \left(1 + z \cdot z\right)\right)\right)}^{3}}}\]
Alternative 24
Error26.9
Cost13568
\[\sqrt[3]{{\left(\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\right)}^{3}}\]
Alternative 25
Error25.8
Cost13568
\[\frac{\frac{1}{x}}{\sqrt[3]{{\left(y \cdot \left(1 + z \cdot z\right)\right)}^{3}}}\]
Alternative 26
Error37.0
Cost13504
\[\frac{\frac{1}{x}}{e^{\log \left(y \cdot \left(1 + z \cdot z\right)\right)}}\]
Alternative 27
Error29.7
Cost7936
\[\frac{\frac{1}{x}}{\frac{y \cdot \left(1 + {z}^{6}\right)}{1 + \left(\left(z \cdot z\right) \cdot \left(z \cdot z\right) - z \cdot z\right)}}\]
Alternative 28
Error24.1
Cost7424
\[\frac{\frac{1}{x}}{\frac{y \cdot \left(1 - {z}^{4}\right)}{1 - z \cdot z}}\]
Alternative 29
Error38.7
Cost7296
\[\frac{\frac{1}{x}}{\left(y \cdot \sqrt{1 + z \cdot z}\right) \cdot \left(-z\right)}\]
Alternative 30
Error6.7
Cost832
\[\frac{1}{x} \cdot \frac{1}{y \cdot \left(1 + z \cdot z\right)}\]
Alternative 31
Error6.5
Cost832
\[\frac{1}{y} \cdot \frac{\frac{1}{x}}{1 + z \cdot z}\]
Alternative 32
Error7.1
Cost704
\[\frac{\frac{\frac{1}{y}}{x}}{1 + z \cdot z}\]
Alternative 33
Error6.7
Cost704
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
Alternative 34
Error6.7
Cost704
\[\frac{\frac{1}{x}}{y + y \cdot \left(z \cdot z\right)}\]
Alternative 35
Error6.9
Cost704
\[\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}\]
Alternative 36
Error7.1
Cost704
\[\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\]
Alternative 37
Error6.7
Cost704
\[\frac{\frac{1}{y \cdot \left(1 + z \cdot z\right)}}{x}\]
Alternative 38
Error6.8
Cost704
\[\frac{1}{y \cdot \left(x \cdot \left(1 + z \cdot z\right)\right)}\]
Alternative 39
Error34.8
Cost576
\[\frac{1}{y \cdot \left(x \cdot \left(z \cdot z\right)\right)}\]
Alternative 40
Error34.9
Cost576
\[\frac{\frac{1}{x}}{y \cdot \left(z \cdot z\right)}\]
Alternative 41
Error33.9
Cost576
\[\frac{1}{\left(z \cdot z\right) \cdot \left(y \cdot x\right)}\]
Alternative 42
Error29.8
Cost448
\[\frac{1}{y} \cdot \frac{1}{x}\]
Alternative 43
Error29.7
Cost320
\[\frac{\frac{1}{x}}{y}\]
Alternative 44
Error29.7
Cost320
\[\frac{1}{y \cdot x}\]
Alternative 45
Error29.7
Cost320
\[\frac{\frac{1}{y}}{x}\]
Alternative 46
Error61.7
Cost64
\[1\]
Alternative 47
Error40.5
Cost64
\[0\]
Alternative 48
Error61.8
Cost64
\[-1\]

Error

Derivation

  1. Initial program 6.7

    \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity_binary64_117266.7

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

  4. Applied add-sqr-sqrt_binary64_117486.7

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

  5. Applied times-frac_binary64_117326.7

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

  6. Applied times-frac_binary64_117326.5

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

  7. Simplified6.5

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

  8. Simplified6.5

    \[\leadsto \frac{1}{y} \cdot \color{blue}{\frac{\frac{1}{x}}{1 + z \cdot z}}\]
  9. Using strategy rm

  10. Applied associate-*l/_binary64_116696.5

    \[\leadsto \color{blue}{\frac{1 \cdot \frac{\frac{1}{x}}{1 + z \cdot z}}{y}}\]

  11. Simplified6.5

    \[\leadsto \frac{\color{blue}{\frac{\frac{1}{x}}{1 + z \cdot z}}}{y}\]

  12. Simplified6.5

    \[\leadsto \color{blue}{\frac{\frac{\frac{1}{x}}{1 + z \cdot z}}{y}}\]

  13. Final simplification6.5

    \[\leadsto \frac{\frac{\frac{1}{x}}{1 + z \cdot z}}{y}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< (* y (+ 1.0 (* z z))) (- INFINITY)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x)) (if (< (* y (+ 1.0 (* z z))) 8.680743250567252e+305) (/ (/ 1.0 x) (* (+ 1.0 (* z z)) y)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x))))

  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))