Average Error: 18.0 → 1.5
Time: 12.7s
Precision: binary64
Cost: 960
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
↓
\[\frac{1}{\frac{-1 - \frac{u}{t1}}{v}} \cdot \frac{1}{u + t1} \]
(FPCore (u v t1) :precision binary64 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))
↓
(FPCore (u v t1)
 :precision binary64
 (* (/ 1.0 (/ (- -1.0 (/ u t1)) v)) (/ 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 (1.0 / ((-1.0 - (u / t1)) / v)) * (1.0 / (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 = (1.0d0 / (((-1.0d0) - (u / t1)) / v)) * (1.0d0 / (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 (1.0 / ((-1.0 - (u / t1)) / v)) * (1.0 / (u + t1));
}

def code(u, v, t1):
	return (-t1 * v) / ((t1 + u) * (t1 + u))
↓
def code(u, v, t1):
	return (1.0 / ((-1.0 - (u / t1)) / v)) * (1.0 / (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(1.0 / Float64(Float64(-1.0 - Float64(u / t1)) / v)) * Float64(1.0 / 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 = (1.0 / ((-1.0 - (u / t1)) / v)) * (1.0 / (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[(1.0 / N[(N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 18.0

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

  2. Simplified1.3

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

    [Start]18.0

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

    *-commutative [=>]18.0

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

    associate-/l* [=>]15.6

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

    associate-*r/ [<=]3.3

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

    associate-/r* [=>]1.4

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

    neg-mul-1 [=>]1.4

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

    associate-/l/ [<=]1.4

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

    metadata-eval [<=]1.4

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

    mul0-lft [<=]8.8

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

    associate-*r/ [=>]1.4

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

    mul0-lft [=>]1.4

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

    *-inverses [<=]1.4

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

    div-sub [<=]1.4

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

    neg-sub0 [<=]1.4

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

    neg-mul-1 [=>]1.4

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

    *-commutative [=>]1.4

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

    associate-/l* [=>]1.4

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

    associate-/l* [<=]1.4

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

    *-commutative [=>]1.4

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

    times-frac [<=]15.5

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

    neg-mul-1 [<=]15.5

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

    associate-/l* [=>]1.4

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

    associate-/r/ [=>]1.4

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

    neg-mul-1 [=>]1.4

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

    *-commutative [=>]1.4

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

    associate-/r* [=>]1.4

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

    *-inverses [=>]1.4

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

    metadata-eval [=>]1.4

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

    neg-mul-1 [<=]1.4

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

    distribute-neg-in [=>]1.4

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

    sub-neg [<=]1.4

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

    div-sub [=>]1.3

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

    neg-sub0 [=>]1.3

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

    div-sub [=>]1.3

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

    mul0-lft [<=]1.3

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

    associate-*r/ [<=]8.8

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

    mul0-lft [=>]1.3

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

    *-inverses [=>]1.3

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

    metadata-eval [=>]1.3

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

  3. Applied egg-rr1.5

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

  4. Final simplification1.5

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

Alternatives

Alternative 1
Error15.1
Cost1437
\[\begin{array}{l} t_1 := \frac{t1}{u} \cdot \frac{-v}{u}\\ t_2 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -2.6 \cdot 10^{-43}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq -9.6 \cdot 10^{-70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -4.7 \cdot 10^{-82}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq 6.6 \cdot 10^{-188}:\\ \;\;\;\;\frac{v}{u \cdot \frac{-u}{t1}}\\ \mathbf{elif}\;t1 \leq 5.2 \cdot 10^{-83} \lor \neg \left(t1 \leq 5.3 \cdot 10^{-17}\right) \land t1 \leq 7.2 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 2
Error15.1
Cost1437
\[\begin{array}{l} t_1 := \frac{t1}{u} \cdot \frac{-v}{u}\\ t_2 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -6.2 \cdot 10^{-44}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq -1.9 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -5.6 \cdot 10^{-82}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq 1.7 \cdot 10^{-188}:\\ \;\;\;\;\frac{v}{u \cdot \frac{-u}{t1}}\\ \mathbf{elif}\;t1 \leq 5.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{-t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{elif}\;t1 \leq 3.8 \cdot 10^{-14} \lor \neg \left(t1 \leq 7.2 \cdot 10^{+103}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 3
Error15.5
Cost1437
\[\begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -6.4 \cdot 10^{-46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -9.6 \cdot 10^{-70}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{elif}\;t1 \leq -4.9 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 1.55 \cdot 10^{-298}:\\ \;\;\;\;\frac{v}{u \cdot \frac{-u}{t1}}\\ \mathbf{elif}\;t1 \leq 3.4 \cdot 10^{-94}:\\ \;\;\;\;\frac{-t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{elif}\;t1 \leq 7.2 \cdot 10^{-8} \lor \neg \left(t1 \leq 7.2 \cdot 10^{+103}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{u}}{\frac{-u}{v}}\\ \end{array} \]
Alternative 4
Error15.4
Cost1437
\[\begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -8.8 \cdot 10^{-44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -9.6 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{v}{u}}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq -2.5 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 2.95 \cdot 10^{-298}:\\ \;\;\;\;\frac{v}{u \cdot \frac{-u}{t1}}\\ \mathbf{elif}\;t1 \leq 5.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{-t1}{\frac{u}{\frac{v}{u}}}\\ \mathbf{elif}\;t1 \leq 8.5 \cdot 10^{-16} \lor \neg \left(t1 \leq 1.2 \cdot 10^{+104}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{u}}{\frac{-u}{v}}\\ \end{array} \]
Alternative 5
Error15.2
Cost1041
\[\begin{array}{l} t_1 := \frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{if}\;u \leq -2.5 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -1.3 \cdot 10^{-57}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq -5.5 \cdot 10^{-63} \lor \neg \left(u \leq 4.6 \cdot 10^{-32}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]
Alternative 6
Error14.4
Cost969
\[\begin{array}{l} \mathbf{if}\;u \leq -5 \cdot 10^{-63} \lor \neg \left(u \leq 5.7 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{t1}{\frac{u + t1}{v} \cdot \left(t1 - u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]
Alternative 7
Error3.2
Cost836
\[\begin{array}{l} t_1 := -1 - \frac{u}{t1}\\ \mathbf{if}\;u \leq 7.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{v}{t_1 \cdot \left(u + t1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u}}{t_1}\\ \end{array} \]
Alternative 8
Error1.5
Cost832
\[\frac{\frac{1}{u + t1}}{\frac{-1 - \frac{u}{t1}}{v}} \]
Alternative 9
Error21.2
Cost713
\[\begin{array}{l} \mathbf{if}\;u \leq -2.1 \cdot 10^{+126} \lor \neg \left(u \leq 5.4 \cdot 10^{+79}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
Alternative 10
Error21.0
Cost712
\[\begin{array}{l} \mathbf{if}\;u \leq -2.65 \cdot 10^{+125}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{elif}\;u \leq 4.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \end{array} \]
Alternative 11
Error20.3
Cost712
\[\begin{array}{l} \mathbf{if}\;u \leq -4.8 \cdot 10^{+125}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{elif}\;u \leq 1.82 \cdot 10^{+78}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \end{array} \]
Alternative 12
Error1.3
Cost704
\[\frac{\frac{v}{u + t1}}{-1 - \frac{u}{t1}} \]
Alternative 13
Error27.6
Cost521
\[\begin{array}{l} \mathbf{if}\;u \leq -1.65 \cdot 10^{+211} \lor \neg \left(u \leq 1.25 \cdot 10^{+122}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]
Alternative 14
Error30.3
Cost256
\[\frac{-v}{t1} \]
Alternative 15
Error54.1
Cost192
\[\frac{v}{t1} \]

Error

Reproduce

herbie shell --seed 2022364 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))