Average Error: 18.6 → 1.2
Time: 6.6s
Precision: binary64
Cost: 768
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\frac{\frac{v}{t1 + u} \cdot \left(-t1\right)}{t1 + u}\]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\frac{\frac{v}{t1 + u} \cdot \left(-t1\right)}{t1 + u}
(FPCore (u v t1) :precision binary64 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))
↓
(FPCore (u v t1) :precision binary64 (/ (* (/ v (+ t1 u)) (- t1)) (+ t1 u)))
double code(double u, double v, double t1) {
	return (-t1 * v) / ((t1 + u) * (t1 + u));
}
↓
double code(double u, double v, double t1) {
	return ((v / (t1 + u)) * -t1) / (t1 + u);
}

Error

Bits error versus u

Bits error versus v

Bits error versus t1

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error56.4
Cost46400
\[\frac{\frac{\sqrt[3]{v} \cdot \sqrt[3]{v}}{\sqrt{t1 + u}}}{\sqrt{-1 - \frac{u}{t1}}} \cdot \frac{\frac{\sqrt[3]{v}}{\sqrt{t1 + u}}}{\sqrt{-1 - \frac{u}{t1}}}\]
Alternative 2
Error49.1
Cost33856
\[\frac{\sqrt[3]{\frac{v}{t1 + u}} \cdot \sqrt[3]{\frac{v}{t1 + u}}}{\sqrt{-1 - \frac{u}{t1}}} \cdot \frac{\sqrt[3]{\frac{v}{t1 + u}}}{\sqrt{-1 - \frac{u}{t1}}}\]
Alternative 3
Error33.1
Cost33344
\[\frac{\sqrt{v}}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}} \cdot \frac{\frac{\sqrt{v}}{\sqrt[3]{t1 + u}}}{-1 - \frac{u}{t1}}\]
Alternative 4
Error32.7
Cost33216
\[\frac{\sqrt[3]{v} \cdot \sqrt[3]{v}}{\sqrt{t1 + u}} \cdot \frac{\frac{\sqrt[3]{v}}{\sqrt{t1 + u}}}{-1 - \frac{u}{t1}}\]
Alternative 5
Error56.3
Cost26816
\[\frac{\sqrt{v}}{\sqrt{-1 - \frac{u}{t1}}} \cdot \frac{\frac{\sqrt{v}}{t1 + u}}{\sqrt{-1 - \frac{u}{t1}}}\]
Alternative 6
Error48.5
Cost26688
\[\frac{\sqrt{v}}{\sqrt{t1 + u}} \cdot \frac{\frac{\sqrt{v}}{\sqrt{t1 + u}}}{-1 - \frac{u}{t1}}\]
Alternative 7
Error2.2
Cost21440
\[\sqrt[3]{\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}} \cdot \left(\sqrt[3]{\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}} \cdot \sqrt[3]{\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}}\right)\]
Alternative 8
Error2.2
Cost20672
\[\left(\sqrt[3]{\frac{v}{t1 + u}} \cdot \sqrt[3]{\frac{v}{t1 + u}}\right) \cdot \frac{\sqrt[3]{\frac{v}{t1 + u}}}{-1 - \frac{u}{t1}}\]
Alternative 9
Error2.2
Cost20672
\[\frac{\sqrt[3]{\frac{v}{t1 + u}} \cdot \sqrt[3]{\frac{v}{t1 + u}}}{\frac{-1 - \frac{u}{t1}}{\sqrt[3]{\frac{v}{t1 + u}}}}\]
Alternative 10
Error2.7
Cost20160
\[\left(\sqrt[3]{v} \cdot \sqrt[3]{v}\right) \cdot \frac{\frac{\sqrt[3]{v}}{t1 + u}}{-1 - \frac{u}{t1}}\]
Alternative 11
Error21.8
Cost14272
\[\sqrt{\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}} \cdot \sqrt{\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}}\]
Alternative 12
Error49.0
Cost14016
\[\frac{1}{\sqrt{-1 - \frac{u}{t1}}} \cdot \frac{\frac{v}{t1 + u}}{\sqrt{-1 - \frac{u}{t1}}}\]
Alternative 13
Error32.6
Cost13888
\[\frac{1}{\sqrt{t1 + u}} \cdot \frac{\frac{v}{\sqrt{t1 + u}}}{-1 - \frac{u}{t1}}\]
Alternative 14
Error33.3
Cost13632
\[\sqrt{v} \cdot \frac{\frac{\sqrt{v}}{t1 + u}}{-1 - \frac{u}{t1}}\]
Alternative 15
Error33.2
Cost13632
\[\frac{\sqrt{v} \cdot \frac{\sqrt{v}}{-1 - \frac{u}{t1}}}{t1 + u}\]
Alternative 16
Error24.1
Cost13568
\[\sqrt[3]{{\left(\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}\right)}^{3}}\]
Alternative 17
Error23.3
Cost8064
\[\frac{\frac{v}{t1 + u}}{-1 - {\left(\frac{u}{t1}\right)}^{3}} \cdot \left(1 + \left(\frac{u}{t1} \cdot \frac{u}{t1} - \frac{u}{t1}\right)\right)\]
Alternative 18
Error23.4
Cost7936
\[\frac{v}{-1 - {\left(\frac{u}{t1}\right)}^{3}} \cdot \frac{1 + \frac{u}{t1} \cdot \left(-1 + \frac{u}{t1}\right)}{t1 + u}\]
Alternative 19
Error13.4
Cost1472
\[\frac{v}{t1 + u} \cdot \left(\left(-1 + \frac{u}{t1}\right) \cdot \frac{1}{1 - \frac{u}{t1} \cdot \frac{u}{t1}}\right)\]
Alternative 20
Error20.9
Cost1216
\[\frac{v}{t1 \cdot t1 - u \cdot u} \cdot \frac{t1 - u}{-1 - \frac{u}{t1}}\]
Alternative 21
Error1.4
Cost832
\[\frac{v}{t1 + u} \cdot \frac{1}{-1 - \frac{u}{t1}}\]
Alternative 22
Error3.3
Cost832
\[v \cdot \frac{\frac{1}{t1 + u}}{-1 - \frac{u}{t1}}\]
Alternative 23
Error1.9
Cost832
\[\frac{1}{\frac{-1 - \frac{u}{t1}}{\frac{v}{t1 + u}}}\]
Alternative 24
Error18.6
Cost768
\[\frac{v \cdot \left(-t1\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
Alternative 25
Error3.4
Cost704
\[\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}\]
Alternative 26
Error1.2
Cost704
\[\frac{\frac{v}{-1 - \frac{u}{t1}}}{t1 + u}\]
Alternative 27
Error1.4
Cost704
\[\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}\]
Alternative 28
Error31.4
Cost640
\[\frac{\frac{-v}{\frac{u}{t1}}}{t1 + u}\]
Alternative 29
Error29.1
Cost640
\[\frac{v}{t1 + u} \cdot \frac{-t1}{u}\]
Alternative 30
Error34.3
Cost512
\[\frac{-v}{u \cdot \frac{u}{t1}}\]
Alternative 31
Error36.4
Cost512
\[\frac{v \cdot \left(-t1\right)}{u \cdot u}\]
Alternative 32
Error24.6
Cost384
\[\frac{-v}{t1 + u}\]
Alternative 33
Error30.4
Cost256
\[-\frac{v}{t1}\]
Alternative 34
Error61.8
Cost64
\[1\]
Alternative 35
Error39.0
Cost64
\[0\]
Alternative 36
Error61.7
Cost64
\[-1\]

Error

Derivation

  1. Initial program 18.6

    \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
  2. Using strategy rm

  3. Applied associate-/r*_binary64_2211.6

    \[\leadsto \color{blue}{\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}}\]

  4. Simplified1.2

    \[\leadsto \frac{\color{blue}{\left(-t1\right) \cdot \frac{v}{t1 + u}}}{t1 + u}\]

  5. Simplified1.2

    \[\leadsto \color{blue}{\frac{\frac{v}{t1 + u} \cdot \left(-t1\right)}{t1 + u}}\]

  6. Final simplification1.2

    \[\leadsto \frac{\frac{v}{t1 + u} \cdot \left(-t1\right)}{t1 + u}\]

Reproduce

herbie shell --seed 2021022 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))