Rosa's DopplerBench

Average Error: 18.4 → 1.6

Time: 13.5s

Precision: binary64

Cost: 1032

Math

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

↓

\[\begin{array}{l} t_1 := \frac{v}{t1 + u}\\ \mathbf{if}\;t1 \leq -4.5 \cdot 10^{-250}:\\ \;\;\;\;\frac{t_1}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 9.5 \cdot 10^{-230}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-t1}{t1 + u}\\ \end{array} \]

FPCore

(FPCore (u v t1) :precision binary64 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))

↓

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ v (+ t1 u))))
   (if (<= t1 -4.5e-250)
     (/ t_1 (- -1.0 (/ u t1)))
     (if (<= t1 9.5e-230)
       (/ (* v (/ t1 u)) (- u))
       (* t_1 (/ (- t1) (+ t1 u)))))))

C

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

↓

double code(double u, double v, double t1) {
	double t_1 = v / (t1 + u);
	double tmp;
	if (t1 <= -4.5e-250) {
		tmp = t_1 / (-1.0 - (u / t1));
	} else if (t1 <= 9.5e-230) {
		tmp = (v * (t1 / u)) / -u;
	} else {
		tmp = t_1 * (-t1 / (t1 + u));
	}
	return tmp;
}

Fortran

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = v / (t1 + u)
    if (t1 <= (-4.5d-250)) then
        tmp = t_1 / ((-1.0d0) - (u / t1))
    else if (t1 <= 9.5d-230) then
        tmp = (v * (t1 / u)) / -u
    else
        tmp = t_1 * (-t1 / (t1 + u))
    end if
    code = tmp
end function

Java

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) {
	double t_1 = v / (t1 + u);
	double tmp;
	if (t1 <= -4.5e-250) {
		tmp = t_1 / (-1.0 - (u / t1));
	} else if (t1 <= 9.5e-230) {
		tmp = (v * (t1 / u)) / -u;
	} else {
		tmp = t_1 * (-t1 / (t1 + u));
	}
	return tmp;
}

Python

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

↓

def code(u, v, t1):
	t_1 = v / (t1 + u)
	tmp = 0
	if t1 <= -4.5e-250:
		tmp = t_1 / (-1.0 - (u / t1))
	elif t1 <= 9.5e-230:
		tmp = (v * (t1 / u)) / -u
	else:
		tmp = t_1 * (-t1 / (t1 + u))
	return tmp

Julia

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

↓

function code(u, v, t1)
	t_1 = Float64(v / Float64(t1 + u))
	tmp = 0.0
	if (t1 <= -4.5e-250)
		tmp = Float64(t_1 / Float64(-1.0 - Float64(u / t1)));
	elseif (t1 <= 9.5e-230)
		tmp = Float64(Float64(v * Float64(t1 / u)) / Float64(-u));
	else
		tmp = Float64(t_1 * Float64(Float64(-t1) / Float64(t1 + u)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(u, v, t1)
	t_1 = v / (t1 + u);
	tmp = 0.0;
	if (t1 <= -4.5e-250)
		tmp = t_1 / (-1.0 - (u / t1));
	elseif (t1 <= 9.5e-230)
		tmp = (v * (t1 / u)) / -u;
	else
		tmp = t_1 * (-t1 / (t1 + u));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[u_, v_, t1_] := Block[{t$95$1 = N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t1, -4.5e-250], N[(t$95$1 / N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 9.5e-230], N[(N[(v * N[(t1 / u), $MachinePrecision]), $MachinePrecision] / (-u)), $MachinePrecision], N[(t$95$1 * N[((-t1) / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -4.49999999999999993e-250

   1. Initial program 18.6

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

   2. Simplified 0.7

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

      [Start] 18.6	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      *-commutative [=>] 18.6	\[ \frac{\color{blue}{v \cdot \left(-t1\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      associate-/l* [=>] 16.2	\[ \color{blue}{\frac{v}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}} \]
      associate-*r/ [<=] 3.5	\[ \frac{v}{\color{blue}{\left(t1 + u\right) \cdot \frac{t1 + u}{-t1}}} \]
      associate-/r* [=>] 0.7	\[ \color{blue}{\frac{\frac{v}{t1 + u}}{\frac{t1 + u}{-t1}}} \]
      neg-mul-1 [=>] 0.7	\[ \frac{\frac{v}{t1 + u}}{\frac{t1 + u}{\color{blue}{-1 \cdot t1}}} \]
      associate-/l/ [<=] 0.7	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{\frac{t1 + u}{t1}}{-1}}} \]
      metadata-eval [<=] 0.7	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{0 - 1}}} \]
      mul0-lft [<=] 6.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{0 \cdot \frac{t1 + u}{t1}} - 1}} \]
      associate-*r/ [=>] 0.7	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{0 \cdot \left(t1 + u\right)}{t1}} - 1}} \]
      mul0-lft [=>] 0.7	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{0}}{t1} - 1}} \]
      *-inverses [<=] 0.7	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{0}{t1} - \color{blue}{\frac{t1}{t1}}}} \]
      div-sub [<=] 0.7	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{0 - t1}{t1}}}} \]
      neg-sub0 [<=] 0.7	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{-t1}}{t1}}} \]
      neg-mul-1 [=>] 0.7	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{-1 \cdot t1}}{t1}}} \]
      *-commutative [=>] 0.7	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{t1 \cdot -1}}{t1}}} \]
      associate-/l* [=>] 0.7	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{t1}{\frac{t1}{-1}}}}} \]
      associate-/l* [<=] 0.7	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{\frac{t1 + u}{t1} \cdot \frac{t1}{-1}}{t1}}} \]
      *-commutative [=>] 0.7	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1}{-1} \cdot \frac{t1 + u}{t1}}}{t1}} \]
      times-frac [<=] 14.6	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1 \cdot \left(t1 + u\right)}{-1 \cdot t1}}}{t1}} \]
      neg-mul-1 [<=] 14.6	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 \cdot \left(t1 + u\right)}{\color{blue}{-t1}}}{t1}} \]
      associate-/l* [=>] 0.8	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1}{\frac{-t1}{t1 + u}}}}{t1}} \]

   if -4.49999999999999993e-250 < t1 < 9.5000000000000004e-230

   1. Initial program 16.5

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

   2. Simplified 6.1

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

      [Start] 16.5	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      *-commutative [=>] 16.5	\[ \frac{\color{blue}{v \cdot \left(-t1\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      associate-/l* [=>] 15.8	\[ \color{blue}{\frac{v}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}} \]
      associate-*r/ [<=] 3.7	\[ \frac{v}{\color{blue}{\left(t1 + u\right) \cdot \frac{t1 + u}{-t1}}} \]
      associate-/r* [=>] 6.1	\[ \color{blue}{\frac{\frac{v}{t1 + u}}{\frac{t1 + u}{-t1}}} \]
      neg-mul-1 [=>] 6.1	\[ \frac{\frac{v}{t1 + u}}{\frac{t1 + u}{\color{blue}{-1 \cdot t1}}} \]
      associate-/l/ [<=] 6.1	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{\frac{t1 + u}{t1}}{-1}}} \]
      metadata-eval [<=] 6.1	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{0 - 1}}} \]
      mul0-lft [<=] 35.0	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{0 \cdot \frac{t1 + u}{t1}} - 1}} \]
      associate-*r/ [=>] 6.1	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{0 \cdot \left(t1 + u\right)}{t1}} - 1}} \]
      mul0-lft [=>] 6.1	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{0}}{t1} - 1}} \]
      *-inverses [<=] 6.1	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{0}{t1} - \color{blue}{\frac{t1}{t1}}}} \]
      div-sub [<=] 6.1	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{0 - t1}{t1}}}} \]
      neg-sub0 [<=] 6.1	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{-t1}}{t1}}} \]
      neg-mul-1 [=>] 6.1	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{-1 \cdot t1}}{t1}}} \]
      *-commutative [=>] 6.1	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{t1 \cdot -1}}{t1}}} \]
      associate-/l* [=>] 6.1	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{t1}{\frac{t1}{-1}}}}} \]
      associate-/l* [<=] 6.1	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{\frac{t1 + u}{t1} \cdot \frac{t1}{-1}}{t1}}} \]
      *-commutative [=>] 6.1	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1}{-1} \cdot \frac{t1 + u}{t1}}}{t1}} \]
      times-frac [<=] 27.2	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1 \cdot \left(t1 + u\right)}{-1 \cdot t1}}}{t1}} \]
      neg-mul-1 [<=] 27.2	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 \cdot \left(t1 + u\right)}{\color{blue}{-t1}}}{t1}} \]
      associate-/l* [=>] 6.1	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1}{\frac{-t1}{t1 + u}}}}{t1}} \]

   3. Taylor expanded in t1 around 0 16.5

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]

   4. Simplified 16.5

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

      [Start] 16.5	\[ -1 \cdot \frac{t1 \cdot v}{{u}^{2}} \]
      associate-*r/ [=>] 16.5	\[ \color{blue}{\frac{-1 \cdot \left(t1 \cdot v\right)}{{u}^{2}}} \]
      associate-*r* [=>] 16.5	\[ \frac{\color{blue}{\left(-1 \cdot t1\right) \cdot v}}{{u}^{2}} \]
      neg-mul-1 [<=] 16.5	\[ \frac{\color{blue}{\left(-t1\right)} \cdot v}{{u}^{2}} \]
      unpow2 [=>] 16.5	\[ \frac{\left(-t1\right) \cdot v}{\color{blue}{u \cdot u}} \]

   5. Taylor expanded in t1 around 0 16.5

      \[\leadsto \color{blue}{-1 \cdot \frac{t1 \cdot v}{{u}^{2}}} \]

   6. Simplified 12.5

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

      [Start] 16.5	\[ -1 \cdot \frac{t1 \cdot v}{{u}^{2}} \]
      unpow2 [=>] 16.5	\[ -1 \cdot \frac{t1 \cdot v}{\color{blue}{u \cdot u}} \]
      associate-*r/ [=>] 16.5	\[ \color{blue}{\frac{-1 \cdot \left(t1 \cdot v\right)}{u \cdot u}} \]
      associate-*l/ [<=] 16.8	\[ \color{blue}{\frac{-1}{u \cdot u} \cdot \left(t1 \cdot v\right)} \]
      associate-*r* [=>] 16.3	\[ \color{blue}{\left(\frac{-1}{u \cdot u} \cdot t1\right) \cdot v} \]
      associate-/r/ [<=] 15.8	\[ \color{blue}{\frac{-1}{\frac{u \cdot u}{t1}}} \cdot v \]
      associate-*l/ [=>] 15.8	\[ \color{blue}{\frac{-1 \cdot v}{\frac{u \cdot u}{t1}}} \]
      neg-mul-1 [<=] 15.8	\[ \frac{\color{blue}{-v}}{\frac{u \cdot u}{t1}} \]
      associate-/r/ [=>] 16.6	\[ \color{blue}{\frac{-v}{u \cdot u} \cdot t1} \]
      *-commutative [=>] 16.6	\[ \color{blue}{t1 \cdot \frac{-v}{u \cdot u}} \]
      associate-/r* [=>] 12.5	\[ t1 \cdot \color{blue}{\frac{\frac{-v}{u}}{u}} \]

   7. Applied egg-rr 12.3

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

   8. Applied egg-rr 9.5

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

   if 9.5000000000000004e-230 < t1

   1. Initial program 18.6

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

   2. Simplified 0.8

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

      [Start] 18.6	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      times-frac [=>] 0.8	\[ \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -4.5 \cdot 10^{-250}:\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 9.5 \cdot 10^{-230}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{t1 + u}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.3
Cost	2441

\[\begin{array}{l} t_1 := \frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{-194} \lor \neg \left(t_1 \leq 10^{+218}\right):\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	14.5
Cost	1040

\[\begin{array}{l} t_1 := \frac{t1}{u} \cdot \frac{-v}{u}\\ t_2 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -1.35 \cdot 10^{-7}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq -8.5 \cdot 10^{-247}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 1.2 \cdot 10^{-241}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{elif}\;t1 \leq 3.4 \cdot 10^{-49}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	1.7
Cost	969

\[\begin{array}{l} \mathbf{if}\;t1 \leq -2 \cdot 10^{-247} \lor \neg \left(t1 \leq 3.9 \cdot 10^{-241}\right):\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \end{array} \]

Alternative 4
Error	14.8
Cost	841

\[\begin{array}{l} \mathbf{if}\;t1 \leq -52 \lor \neg \left(t1 \leq 3.9 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\\ \end{array} \]

Alternative 5
Error	14.7
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -0.0034 \lor \neg \left(t1 \leq 7.8 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{u}}{u}\\ \end{array} \]

Alternative 6
Error	13.8
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -0.00086 \lor \neg \left(t1 \leq 1.5 \cdot 10^{-49}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \]

Alternative 7
Error	14.7
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -1.85 \cdot 10^{-8} \lor \neg \left(t1 \leq 1.02 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{\frac{u}{\frac{t1}{u}}}\\ \end{array} \]

Alternative 8
Error	20.8
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -3.4 \cdot 10^{+79} \lor \neg \left(u \leq 3.6 \cdot 10^{+69}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 9
Error	21.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+78}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{elif}\;u \leq 5.1 \cdot 10^{+70}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \end{array} \]

Alternative 10
Error	20.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;u \leq -1.02 \cdot 10^{+99}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{elif}\;u \leq 1.75 \cdot 10^{+98}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \end{array} \]

Alternative 11
Error	27.8
Cost	520

\[\begin{array}{l} \mathbf{if}\;u \leq -5.6 \cdot 10^{+79}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 1.4 \cdot 10^{+141}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 12
Error	50.5
Cost	456

\[\begin{array}{l} \mathbf{if}\;u \leq -3.45 \cdot 10^{+78}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 7.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 13
Error	54.8
Cost	192

\[\frac{v}{t1} \]

Reproduce

herbie shell --seed 2023032 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))