Average Error: 18.3 → 1.4
Time: 12.5s
Precision: binary64
Cost: 768
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
↓
\[\frac{\frac{v}{1 + \frac{u}{t1}}}{\left(-u\right) - t1} \]
(FPCore (u v t1) :precision binary64 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))
↓
(FPCore (u v t1) :precision binary64 (/ (/ v (+ 1.0 (/ u t1))) (- (- 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 / (1.0 + (u / t1))) / (-u - t1);
}

real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (-t1 * v) / ((t1 + u) * (t1 + u))
end function
↓
real(8) function code(u, v, t1)
    real(8), intent (in) :: u
    real(8), intent (in) :: v
    real(8), intent (in) :: t1
    code = (v / (1.0d0 + (u / t1))) / (-u - t1)
end function

public static double code(double u, double v, double t1) {
	return (-t1 * v) / ((t1 + u) * (t1 + u));
}
↓
public static double code(double u, double v, double t1) {
	return (v / (1.0 + (u / t1))) / (-u - t1);
}

def code(u, v, t1):
	return (-t1 * v) / ((t1 + u) * (t1 + u))
↓
def code(u, v, t1):
	return (v / (1.0 + (u / t1))) / (-u - t1)

function code(u, v, t1)
	return Float64(Float64(Float64(-t1) * v) / Float64(Float64(t1 + u) * Float64(t1 + u)))
end
↓
function code(u, v, t1)
	return Float64(Float64(v / Float64(1.0 + Float64(u / t1))) / Float64(Float64(-u) - t1))
end

function tmp = code(u, v, t1)
	tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
end
↓
function tmp = code(u, v, t1)
	tmp = (v / (1.0 + (u / t1))) / (-u - t1);
end

code[u_, v_, t1_] := N[(N[((-t1) * v), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[u_, v_, t1_] := N[(N[(v / N[(1.0 + N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-u) - t1), $MachinePrecision]), $MachinePrecision]

\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\frac{\frac{v}{1 + \frac{u}{t1}}}{\left(-u\right) - t1}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 18.3

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

  2. Simplified1.5

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

    [Start]18.3

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

    *-commutative [=>]18.3

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

    associate-/l* [=>]15.9

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

    associate-*r/ [<=]3.3

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

    associate-/r* [=>]1.5

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

    neg-mul-1 [=>]1.5

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

    associate-/l/ [<=]1.5

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

    metadata-eval [<=]1.5

    \[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{0 - 1}}} \]

    mul0-lft [<=]8.9

    \[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{0 \cdot \frac{t1 + u}{t1}} - 1}} \]

    associate-*r/ [=>]1.5

    \[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{0 \cdot \left(t1 + u\right)}{t1}} - 1}} \]

    mul0-lft [=>]1.5

    \[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{0}}{t1} - 1}} \]

    *-inverses [<=]1.5

    \[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{0}{t1} - \color{blue}{\frac{t1}{t1}}}} \]

    div-sub [<=]1.5

    \[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{0 - t1}{t1}}}} \]

    neg-sub0 [<=]1.5

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

    neg-mul-1 [=>]1.5

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

    *-commutative [=>]1.5

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

    associate-/l* [=>]1.5

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

    associate-/l* [<=]1.5

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

    *-commutative [=>]1.5

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

    times-frac [<=]16.3

    \[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1 \cdot \left(t1 + u\right)}{-1 \cdot t1}}}{t1}} \]

    neg-mul-1 [<=]16.3

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

    associate-/l* [=>]1.5

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

    associate-/r/ [=>]1.5

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

    neg-mul-1 [=>]1.5

    \[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1}{\color{blue}{-1 \cdot t1}} \cdot \left(t1 + u\right)}{t1}} \]

    *-commutative [=>]1.5

    \[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1}{\color{blue}{t1 \cdot -1}} \cdot \left(t1 + u\right)}{t1}} \]

    associate-/r* [=>]1.5

    \[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{\frac{t1}{t1}}{-1}} \cdot \left(t1 + u\right)}{t1}} \]

    *-inverses [=>]1.5

    \[ \frac{\frac{v}{t1 + u}}{\frac{\frac{\color{blue}{1}}{-1} \cdot \left(t1 + u\right)}{t1}} \]

    metadata-eval [=>]1.5

    \[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{-1} \cdot \left(t1 + u\right)}{t1}} \]

    neg-mul-1 [<=]1.5

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

    distribute-neg-in [=>]1.5

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

    sub-neg [<=]1.5

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

    div-sub [=>]1.5

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

    neg-sub0 [=>]1.5

    \[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - t1}}{t1} - \frac{u}{t1}} \]

    div-sub [=>]1.5

    \[ \frac{\frac{v}{t1 + u}}{\color{blue}{\left(\frac{0}{t1} - \frac{t1}{t1}\right)} - \frac{u}{t1}} \]

    mul0-lft [<=]1.5

    \[ \frac{\frac{v}{t1 + u}}{\left(\frac{\color{blue}{0 \cdot \left(t1 + u\right)}}{t1} - \frac{t1}{t1}\right) - \frac{u}{t1}} \]

    associate-*r/ [<=]8.9

    \[ \frac{\frac{v}{t1 + u}}{\left(\color{blue}{0 \cdot \frac{t1 + u}{t1}} - \frac{t1}{t1}\right) - \frac{u}{t1}} \]

    mul0-lft [=>]1.5

    \[ \frac{\frac{v}{t1 + u}}{\left(\color{blue}{0} - \frac{t1}{t1}\right) - \frac{u}{t1}} \]

    *-inverses [=>]1.5

    \[ \frac{\frac{v}{t1 + u}}{\left(0 - \color{blue}{1}\right) - \frac{u}{t1}} \]

    metadata-eval [=>]1.5

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

  3. Applied egg-rr3.2

    \[\leadsto \color{blue}{\frac{\frac{-1}{\frac{u}{t1} + 1}}{\left(-u\right) - t1} \cdot \left(-v\right)} \]

  4. Simplified1.4

    \[\leadsto \color{blue}{\frac{\frac{-v}{-1 - \frac{u}{t1}}}{\left(-u\right) - t1}} \]
    Proof

    [Start]3.2

    \[ \frac{\frac{-1}{\frac{u}{t1} + 1}}{\left(-u\right) - t1} \cdot \left(-v\right) \]

    associate-*l/ [=>]1.4

    \[ \color{blue}{\frac{\frac{-1}{\frac{u}{t1} + 1} \cdot \left(-v\right)}{\left(-u\right) - t1}} \]

    *-commutative [<=]1.4

    \[ \frac{\color{blue}{\left(-v\right) \cdot \frac{-1}{\frac{u}{t1} + 1}}}{\left(-u\right) - t1} \]

    distribute-lft-neg-out [=>]1.4

    \[ \frac{\color{blue}{-v \cdot \frac{-1}{\frac{u}{t1} + 1}}}{\left(-u\right) - t1} \]

    /-rgt-identity [<=]1.4

    \[ \frac{-\color{blue}{\frac{v}{1}} \cdot \frac{-1}{\frac{u}{t1} + 1}}{\left(-u\right) - t1} \]

    associate-/r/ [<=]1.4

    \[ \frac{-\color{blue}{\frac{v}{\frac{1}{\frac{-1}{\frac{u}{t1} + 1}}}}}{\left(-u\right) - t1} \]

    associate-/r/ [=>]1.4

    \[ \frac{-\frac{v}{\color{blue}{\frac{1}{-1} \cdot \left(\frac{u}{t1} + 1\right)}}}{\left(-u\right) - t1} \]

    metadata-eval [=>]1.4

    \[ \frac{-\frac{v}{\color{blue}{-1} \cdot \left(\frac{u}{t1} + 1\right)}}{\left(-u\right) - t1} \]

    mul-1-neg [=>]1.4

    \[ \frac{-\frac{v}{\color{blue}{-\left(\frac{u}{t1} + 1\right)}}}{\left(-u\right) - t1} \]

    +-commutative [<=]1.4

    \[ \frac{-\frac{v}{-\color{blue}{\left(1 + \frac{u}{t1}\right)}}}{\left(-u\right) - t1} \]

    distribute-neg-in [=>]1.4

    \[ \frac{-\frac{v}{\color{blue}{\left(-1\right) + \left(-\frac{u}{t1}\right)}}}{\left(-u\right) - t1} \]

    metadata-eval [=>]1.4

    \[ \frac{-\frac{v}{\color{blue}{-1} + \left(-\frac{u}{t1}\right)}}{\left(-u\right) - t1} \]

    sub-neg [<=]1.4

    \[ \frac{-\frac{v}{\color{blue}{-1 - \frac{u}{t1}}}}{\left(-u\right) - t1} \]

    distribute-neg-frac [=>]1.4

    \[ \frac{\color{blue}{\frac{-v}{-1 - \frac{u}{t1}}}}{\left(-u\right) - t1} \]

  5. Taylor expanded in v around 0 1.4

    \[\leadsto \frac{\color{blue}{\frac{v}{1 + \frac{u}{t1}}}}{\left(-u\right) - t1} \]

  6. Final simplification1.4

    \[\leadsto \frac{\frac{v}{1 + \frac{u}{t1}}}{\left(-u\right) - t1} \]

Alternatives

Alternative 1
Error16.1
Cost1305
\[\begin{array}{l} t_1 := \frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{if}\;u \leq -1.95 \cdot 10^{+115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -2 \cdot 10^{+102}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq -780000000000:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{u \cdot u}\\ \mathbf{elif}\;u \leq 1.4 \cdot 10^{-8} \lor \neg \left(u \leq 1.5 \cdot 10^{+78}\right) \land u \leq 5.5 \cdot 10^{+97}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 2
Error15.5
Cost1304
\[\begin{array}{l} t_1 := \frac{-t1}{u \cdot \frac{u}{v}}\\ t_2 := \frac{t1}{u} \cdot \frac{-v}{u}\\ t_3 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;u \leq -9.5 \cdot 10^{+114}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq -1.65 \cdot 10^{+102}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq -10000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 1.5 \cdot 10^{-8}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;u \leq 4.7 \cdot 10^{+79}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq 4.9 \cdot 10^{+95}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 3
Error15.5
Cost1304
\[\begin{array}{l} t_1 := \frac{-t1}{u \cdot \frac{u}{v}}\\ t_2 := \frac{-v}{u}\\ t_3 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;u \leq -9.5 \cdot 10^{+114}:\\ \;\;\;\;\frac{t1 \cdot t_2}{u}\\ \mathbf{elif}\;u \leq -2.15 \cdot 10^{+102}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq -11000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 9.5 \cdot 10^{-9}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;u \leq 1.75 \cdot 10^{+77}:\\ \;\;\;\;\frac{t1}{u} \cdot t_2\\ \mathbf{elif}\;u \leq 4.9 \cdot 10^{+95}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 4
Error15.5
Cost1040
\[\begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;u \leq -5500000000000:\\ \;\;\;\;-\frac{\frac{t1}{u}}{\frac{u}{v}}\\ \mathbf{elif}\;u \leq 1.5 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 1.02 \cdot 10^{+80}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{elif}\;u \leq 4.9 \cdot 10^{+95}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \end{array} \]
Alternative 5
Error15.3
Cost777
\[\begin{array}{l} \mathbf{if}\;t1 \leq -2 \cdot 10^{-58} \lor \neg \left(t1 \leq 4.5 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-v\right) \cdot \frac{t1}{u \cdot u}\\ \end{array} \]
Alternative 6
Error13.3
Cost777
\[\begin{array}{l} \mathbf{if}\;t1 \leq -1.35 \cdot 10^{-58} \lor \neg \left(t1 \leq 6.4 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \end{array} \]
Alternative 7
Error1.4
Cost768
\[\frac{-t1}{u + t1} \cdot \frac{v}{u + t1} \]
Alternative 8
Error22.1
Cost713
\[\begin{array}{l} \mathbf{if}\;t1 \leq -8 \cdot 10^{-59} \lor \neg \left(t1 \leq 2.7 \cdot 10^{-149}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{u}\\ \end{array} \]
Alternative 9
Error22.1
Cost713
\[\begin{array}{l} \mathbf{if}\;t1 \leq -2.7 \cdot 10^{-59} \lor \neg \left(t1 \leq 9.2 \cdot 10^{-150}\right):\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \]
Alternative 10
Error21.9
Cost713
\[\begin{array}{l} \mathbf{if}\;t1 \leq -2.6 \cdot 10^{-59} \lor \neg \left(t1 \leq 1.8 \cdot 10^{-149}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \]
Alternative 11
Error1.5
Cost704
\[\frac{\frac{v}{u + t1}}{-1 - \frac{u}{t1}} \]
Alternative 12
Error27.8
Cost585
\[\begin{array}{l} \mathbf{if}\;u \leq -8 \cdot 10^{+167} \lor \neg \left(u \leq 3.15 \cdot 10^{+233}\right):\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
Alternative 13
Error27.9
Cost521
\[\begin{array}{l} \mathbf{if}\;u \leq -1.9 \cdot 10^{+177} \lor \neg \left(u \leq 3.15 \cdot 10^{+233}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
Alternative 14
Error27.8
Cost520
\[\begin{array}{l} \mathbf{if}\;u \leq -3.6 \cdot 10^{+175}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 2.4 \cdot 10^{+229}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]
Alternative 15
Error25.0
Cost384
\[\frac{v}{\left(-u\right) - t1} \]
Alternative 16
Error53.5
Cost192
\[\frac{v}{u} \]

Error

Reproduce

herbie shell --seed 2022354 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))