Average Error: 18.3 → 1.7
Time: 7.7s
Precision: binary64
Cost: 704
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}\]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}
(FPCore (u v t1) :precision binary64 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))
↓
(FPCore (u v t1) :precision binary64 (/ (/ v (+ t1 u)) (- -1.0 (/ u t1))))
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)) / (-1.0 - (u / t1));
}

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
Error1.9
Cost59840
\[\frac{\frac{\sqrt[3]{v} \cdot \sqrt[3]{v}}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}}}{\sqrt[3]{-1 - \frac{u}{t1}} \cdot \sqrt[3]{-1 - \frac{u}{t1}}} \cdot \frac{\frac{\sqrt[3]{v}}{\sqrt[3]{t1 + u}}}{\sqrt[3]{-1 - \frac{u}{t1}}}\]
Alternative 2
Error32.7
Cost53312
\[\frac{\frac{\sqrt{v}}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}}}{\sqrt[3]{-1 - \frac{u}{t1}} \cdot \sqrt[3]{-1 - \frac{u}{t1}}} \cdot \frac{\frac{\sqrt{v}}{\sqrt[3]{t1 + u}}}{\sqrt[3]{-1 - \frac{u}{t1}}}\]
Alternative 3
Error2.5
Cost40640
\[\frac{\sqrt[3]{\frac{v}{t1 + u}} \cdot \sqrt[3]{\frac{v}{t1 + u}}}{\sqrt[3]{-1 - \frac{u}{t1}} \cdot \sqrt[3]{-1 - \frac{u}{t1}}} \cdot \frac{\sqrt[3]{\frac{v}{t1 + u}}}{\sqrt[3]{-1 - \frac{u}{t1}}}\]
Alternative 4
Error1.9
Cost39872
\[\frac{\sqrt[3]{v} \cdot \sqrt[3]{v}}{\sqrt[3]{t1 + u} \cdot \sqrt[3]{t1 + u}} \cdot \frac{\frac{\sqrt[3]{v}}{\sqrt[3]{t1 + u}}}{-1 - \frac{u}{t1}}\]
Alternative 5
Error26.5
Cost33856
\[\frac{\sqrt{\frac{v}{t1 + u}}}{\sqrt[3]{-1 - \frac{u}{t1}} \cdot \sqrt[3]{-1 - \frac{u}{t1}}} \cdot \frac{\sqrt{\frac{v}{t1 + u}}}{\sqrt[3]{-1 - \frac{u}{t1}}}\]
Alternative 6
Error48.7
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 7
Error32.6
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 8
Error32.9
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 9
Error54.9
Cost27072
\[\frac{\sqrt{\frac{v}{t1 + u}}}{\sqrt{-1 - \frac{u}{t1}}} \cdot \frac{\sqrt{\frac{v}{t1 + u}}}{\sqrt{-1 - \frac{u}{t1}}}\]
Alternative 10
Error2.4
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 11
Error1.8
Cost20800
\[\frac{v}{\sqrt[3]{-1 - \frac{u}{t1}} \cdot \sqrt[3]{-1 - \frac{u}{t1}}} \cdot \frac{\frac{1}{t1 + u}}{\sqrt[3]{-1 - \frac{u}{t1}}}\]
Alternative 12
Error2.4
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 13
Error2.8
Cost20160
\[\left(\sqrt[3]{v} \cdot \sqrt[3]{v}\right) \cdot \frac{\frac{\sqrt[3]{v}}{t1 + u}}{-1 - \frac{u}{t1}}\]
Alternative 14
Error22.5
Cost14272
\[\sqrt{\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}} \cdot \sqrt{\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}}\]
Alternative 15
Error43.4
Cost14272
\[\frac{v}{{t1}^{3} + {u}^{3}} \cdot \frac{t1 \cdot t1 + u \cdot \left(u - t1\right)}{-1 - \frac{u}{t1}}\]
Alternative 16
Error32.9
Cost13888
\[\frac{1}{\sqrt{t1 + u}} \cdot \frac{\frac{v}{\sqrt{t1 + u}}}{-1 - \frac{u}{t1}}\]
Alternative 17
Error26.4
Cost13888
\[\sqrt{\frac{v}{t1 + u}} \cdot \frac{\sqrt{\frac{v}{t1 + u}}}{-1 - \frac{u}{t1}}\]
Alternative 18
Error33.0
Cost13760
\[\frac{\frac{\frac{v}{\sqrt{t1 + u}}}{\sqrt{t1 + u}}}{-1 - \frac{u}{t1}}\]
Alternative 19
Error25.6
Cost13696
\[v \cdot \sqrt[3]{\frac{1}{{\left(\left(-1 - \frac{u}{t1}\right) \cdot \left(t1 + u\right)\right)}^{3}}}\]
Alternative 20
Error32.7
Cost13632
\[\sqrt{v} \cdot \frac{\frac{\sqrt{v}}{t1 + u}}{-1 - \frac{u}{t1}}\]
Alternative 21
Error24.3
Cost13568
\[\sqrt[3]{{\left(\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}\right)}^{3}}\]
Alternative 22
Error26.0
Cost8576
\[\frac{\frac{v}{t1 + u} \cdot \left(-1 + {\left(\frac{u}{t1}\right)}^{3}\right)}{\left(1 - \frac{u}{t1} \cdot \frac{u}{t1}\right) \cdot \left(1 + \frac{u}{t1} \cdot \left(1 + \frac{u}{t1}\right)\right)}\]
Alternative 23
Error23.5
Cost8192
\[v \cdot \left(\frac{\frac{1}{t1 + u}}{-1 - {\left(\frac{u}{t1}\right)}^{3}} \cdot \left(1 + \left(\frac{u}{t1} \cdot \frac{u}{t1} - \frac{u}{t1}\right)\right)\right)\]
Alternative 24
Error18.6
Cost1344
\[v \cdot \left(\frac{1}{t1 \cdot t1 - u \cdot u} \cdot \frac{t1 - u}{-1 - \frac{u}{t1}}\right)\]
Alternative 25
Error15.1
Cost1344
\[\frac{\frac{v}{t1 + u}}{1 - \frac{u}{t1} \cdot \frac{u}{t1}} \cdot \left(-1 + \frac{u}{t1}\right)\]
Alternative 26
Error14.7
Cost1344
\[\frac{\frac{v}{t1 + u} \cdot \left(-1 + \frac{u}{t1}\right)}{1 - \frac{u}{t1} \cdot \frac{u}{t1}}\]
Alternative 27
Error18.6
Cost1344
\[v \cdot \frac{\frac{1}{t1 \cdot t1 - u \cdot u}}{\frac{-1 - \frac{u}{t1}}{t1 - u}}\]
Alternative 28
Error20.9
Cost1216
\[\frac{t1 - u}{-1 - \frac{u}{t1}} \cdot \frac{v}{t1 \cdot t1 - u \cdot u}\]
Alternative 29
Error3.4
Cost832
\[v \cdot \frac{\frac{1}{t1 + u}}{-1 - \frac{u}{t1}}\]
Alternative 30
Error2.2
Cost832
\[\frac{1}{\frac{-1 - \frac{u}{t1}}{\frac{v}{t1 + u}}}\]
Alternative 31
Error2.0
Cost832
\[\frac{\frac{1}{\frac{t1 + u}{v}}}{-1 - \frac{u}{t1}}\]
Alternative 32
Error3.6
Cost832
\[v \cdot \frac{1}{\left(-1 - \frac{u}{t1}\right) \cdot \left(t1 + u\right)}\]
Alternative 33
Error18.3
Cost768
\[\frac{v \cdot \left(-t1\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
Alternative 34
Error3.5
Cost768
\[\frac{-v}{t1 + u \cdot \left(\frac{u}{t1} + 2\right)}\]
Alternative 35
Error3.5
Cost704
\[\frac{v}{\left(-1 - \frac{u}{t1}\right) \cdot \left(t1 + u\right)}\]
Alternative 36
Error32.8
Cost704
\[v \cdot \frac{\frac{1}{u}}{-1 - \frac{u}{t1}}\]
Alternative 37
Error31.2
Cost576
\[\frac{\frac{v}{u}}{-1 - \frac{u}{t1}}\]
Alternative 38
Error36.1
Cost512
\[v \cdot \left(-\frac{t1}{u \cdot u}\right)\]
Alternative 39
Error25.0
Cost512
\[\frac{-v}{t1 + u \cdot 2}\]
Alternative 40
Error33.9
Cost512
\[\frac{-v}{u \cdot \frac{u}{t1}}\]
Alternative 41
Error30.9
Cost256
\[\frac{-v}{t1}\]
Alternative 42
Error61.8
Cost64
\[1\]
Alternative 43
Error39.7
Cost64
\[0\]
Alternative 44
Error61.7
Cost64
\[-1\]

Error

Derivation

  1. Initial program 18.3

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

  2. Simplified1.7

    \[\leadsto \color{blue}{\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}}\]

  3. Simplified1.7

    \[\leadsto \color{blue}{\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}}\]

  4. Final simplification1.7

    \[\leadsto \frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))