Rosa's DopplerBench

Average Error: 18.2 → 1.3

Time: 15.1s

Precision: binary64

Cost: 969

Math

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

↓

\[\begin{array}{l} t_1 := -1 - \frac{u}{t1}\\ \mathbf{if}\;t1 \leq -3.9 \cdot 10^{-104} \lor \neg \left(t1 \leq 1.04 \cdot 10^{-200}\right):\\ \;\;\;\;\frac{\frac{v}{u + t1}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t_1 \cdot \left(u + t1\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (- -1.0 (/ u t1))))
   (if (or (<= t1 -3.9e-104) (not (<= t1 1.04e-200)))
     (/ (/ v (+ u t1)) t_1)
     (/ v (* t_1 (+ u t1))))))

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 = -1.0 - (u / t1);
	double tmp;
	if ((t1 <= -3.9e-104) || !(t1 <= 1.04e-200)) {
		tmp = (v / (u + t1)) / t_1;
	} else {
		tmp = v / (t_1 * (u + t1));
	}
	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 = (-1.0d0) - (u / t1)
    if ((t1 <= (-3.9d-104)) .or. (.not. (t1 <= 1.04d-200))) then
        tmp = (v / (u + t1)) / t_1
    else
        tmp = v / (t_1 * (u + t1))
    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 = -1.0 - (u / t1);
	double tmp;
	if ((t1 <= -3.9e-104) || !(t1 <= 1.04e-200)) {
		tmp = (v / (u + t1)) / t_1;
	} else {
		tmp = v / (t_1 * (u + t1));
	}
	return tmp;
}

Python

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

↓

def code(u, v, t1):
	t_1 = -1.0 - (u / t1)
	tmp = 0
	if (t1 <= -3.9e-104) or not (t1 <= 1.04e-200):
		tmp = (v / (u + t1)) / t_1
	else:
		tmp = v / (t_1 * (u + t1))
	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(-1.0 - Float64(u / t1))
	tmp = 0.0
	if ((t1 <= -3.9e-104) || !(t1 <= 1.04e-200))
		tmp = Float64(Float64(v / Float64(u + t1)) / t_1);
	else
		tmp = Float64(v / Float64(t_1 * Float64(u + t1)));
	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 = -1.0 - (u / t1);
	tmp = 0.0;
	if ((t1 <= -3.9e-104) || ~((t1 <= 1.04e-200)))
		tmp = (v / (u + t1)) / t_1;
	else
		tmp = v / (t_1 * (u + t1));
	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[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t1, -3.9e-104], N[Not[LessEqual[t1, 1.04e-200]], $MachinePrecision]], N[(N[(v / N[(u + t1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(v / N[(t$95$1 * N[(u + t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t1 < -3.9000000000000002e-104 or 1.03999999999999993e-200 < t1

   1. Initial program 19.3

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

   2. Simplified 0.4

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

      [Start] 19.3	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      *-commutative [=>] 19.3	\[ \frac{\color{blue}{v \cdot \left(-t1\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      associate-/l* [=>] 16.4	\[ \color{blue}{\frac{v}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}} \]
      associate-*r/ [<=] 3.0	\[ \frac{v}{\color{blue}{\left(t1 + u\right) \cdot \frac{t1 + u}{-t1}}} \]
      associate-/r* [=>] 0.4	\[ \color{blue}{\frac{\frac{v}{t1 + u}}{\frac{t1 + u}{-t1}}} \]
      neg-mul-1 [=>] 0.4	\[ \frac{\frac{v}{t1 + u}}{\frac{t1 + u}{\color{blue}{-1 \cdot t1}}} \]
      associate-/l/ [<=] 0.4	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{\frac{t1 + u}{t1}}{-1}}} \]
      metadata-eval [<=] 0.4	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{0 - 1}}} \]
      mul0-lft [<=] 3.3	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{0 \cdot \frac{t1 + u}{t1}} - 1}} \]
      associate-*r/ [=>] 0.4	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{0 \cdot \left(t1 + u\right)}{t1}} - 1}} \]
      mul0-lft [=>] 0.4	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{0}}{t1} - 1}} \]
      *-inverses [<=] 0.4	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{0}{t1} - \color{blue}{\frac{t1}{t1}}}} \]
      div-sub [<=] 0.4	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{0 - t1}{t1}}}} \]
      neg-sub0 [<=] 0.4	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{-t1}}{t1}}} \]
      neg-mul-1 [=>] 0.4	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{-1 \cdot t1}}{t1}}} \]
      *-commutative [=>] 0.4	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{t1 \cdot -1}}{t1}}} \]
      associate-/l* [=>] 0.4	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{t1}{\frac{t1}{-1}}}}} \]
      associate-/l* [<=] 0.4	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{\frac{t1 + u}{t1} \cdot \frac{t1}{-1}}{t1}}} \]
      *-commutative [=>] 0.4	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1}{-1} \cdot \frac{t1 + u}{t1}}}{t1}} \]
      times-frac [<=] 14.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1 \cdot \left(t1 + u\right)}{-1 \cdot t1}}}{t1}} \]
      neg-mul-1 [<=] 14.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 \cdot \left(t1 + u\right)}{\color{blue}{-t1}}}{t1}} \]
      associate-/l* [=>] 0.4	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1}{\frac{-t1}{t1 + u}}}}{t1}} \]

   if -3.9000000000000002e-104 < t1 < 1.03999999999999993e-200

   1. Initial program 15.0

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

   2. Simplified 5.0

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

      [Start] 15.0	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      *-commutative [=>] 15.0	\[ \frac{\color{blue}{v \cdot \left(-t1\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      associate-/l* [=>] 14.6	\[ \color{blue}{\frac{v}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}} \]
      associate-*r/ [<=] 4.2	\[ \frac{v}{\color{blue}{\left(t1 + u\right) \cdot \frac{t1 + u}{-t1}}} \]
      associate-/r* [=>] 5.0	\[ \color{blue}{\frac{\frac{v}{t1 + u}}{\frac{t1 + u}{-t1}}} \]
      neg-mul-1 [=>] 5.0	\[ \frac{\frac{v}{t1 + u}}{\frac{t1 + u}{\color{blue}{-1 \cdot t1}}} \]
      associate-/l/ [<=] 5.0	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{\frac{t1 + u}{t1}}{-1}}} \]
      metadata-eval [<=] 5.0	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{0 - 1}}} \]
      mul0-lft [<=] 28.3	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{0 \cdot \frac{t1 + u}{t1}} - 1}} \]
      associate-*r/ [=>] 5.0	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{0 \cdot \left(t1 + u\right)}{t1}} - 1}} \]
      mul0-lft [=>] 5.0	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{0}}{t1} - 1}} \]
      *-inverses [<=] 5.0	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{0}{t1} - \color{blue}{\frac{t1}{t1}}}} \]
      div-sub [<=] 5.0	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{0 - t1}{t1}}}} \]
      neg-sub0 [<=] 5.0	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{-t1}}{t1}}} \]
      neg-mul-1 [=>] 5.0	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{-1 \cdot t1}}{t1}}} \]
      *-commutative [=>] 5.0	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{t1 \cdot -1}}{t1}}} \]
      associate-/l* [=>] 5.0	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{t1}{\frac{t1}{-1}}}}} \]
      associate-/l* [<=] 5.0	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{\frac{t1 + u}{t1} \cdot \frac{t1}{-1}}{t1}}} \]
      *-commutative [=>] 5.0	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1}{-1} \cdot \frac{t1 + u}{t1}}}{t1}} \]
      times-frac [<=] 20.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1 \cdot \left(t1 + u\right)}{-1 \cdot t1}}}{t1}} \]
      neg-mul-1 [<=] 20.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 \cdot \left(t1 + u\right)}{\color{blue}{-t1}}}{t1}} \]
      associate-/l* [=>] 5.1	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1}{\frac{-t1}{t1 + u}}}}{t1}} \]

   3. Taylor expanded in v around 0 4.2

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

   4. Simplified 4.2

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

      [Start] 4.2	\[ -1 \cdot \frac{v}{\left(t1 + u\right) \cdot \left(1 + \frac{u}{t1}\right)} \]
      associate-*r/ [=>] 4.2	\[ \color{blue}{\frac{-1 \cdot v}{\left(t1 + u\right) \cdot \left(1 + \frac{u}{t1}\right)}} \]
      +-commutative [=>] 4.2	\[ \frac{-1 \cdot v}{\left(t1 + u\right) \cdot \color{blue}{\left(\frac{u}{t1} + 1\right)}} \]
      *-commutative [<=] 4.2	\[ \frac{-1 \cdot v}{\color{blue}{\left(\frac{u}{t1} + 1\right) \cdot \left(t1 + u\right)}} \]
      times-frac [=>] 5.1	\[ \color{blue}{\frac{-1}{\frac{u}{t1} + 1} \cdot \frac{v}{t1 + u}} \]
      /-rgt-identity [<=] 5.1	\[ \color{blue}{\frac{\frac{-1}{\frac{u}{t1} + 1}}{1}} \cdot \frac{v}{t1 + u} \]
      *-commutative [=>] 5.1	\[ \color{blue}{\frac{v}{t1 + u} \cdot \frac{\frac{-1}{\frac{u}{t1} + 1}}{1}} \]
      associate-*r/ [=>] 5.1	\[ \color{blue}{\frac{\frac{v}{t1 + u} \cdot \frac{-1}{\frac{u}{t1} + 1}}{1}} \]
      associate-/l* [=>] 5.1	\[ \color{blue}{\frac{\frac{v}{t1 + u}}{\frac{1}{\frac{-1}{\frac{u}{t1} + 1}}}} \]
      associate-/r/ [=>] 5.0	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{1}{-1} \cdot \left(\frac{u}{t1} + 1\right)}} \]
      metadata-eval [=>] 5.0	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{-1} \cdot \left(\frac{u}{t1} + 1\right)} \]
      mul-1-neg [=>] 5.0	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{-\left(\frac{u}{t1} + 1\right)}} \]
      +-commutative [<=] 5.0	\[ \frac{\frac{v}{t1 + u}}{-\color{blue}{\left(1 + \frac{u}{t1}\right)}} \]
      distribute-neg-in [=>] 5.0	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\left(-1\right) + \left(-\frac{u}{t1}\right)}} \]
      metadata-eval [=>] 5.0	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{-1} + \left(-\frac{u}{t1}\right)} \]
      sub-neg [<=] 5.0	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{-1 - \frac{u}{t1}}} \]
      associate-/r* [<=] 4.2	\[ \color{blue}{\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3.9 \cdot 10^{-104} \lor \neg \left(t1 \leq 1.04 \cdot 10^{-200}\right):\\ \;\;\;\;\frac{\frac{v}{u + t1}}{-1 - \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-1 - \frac{u}{t1}\right) \cdot \left(u + t1\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	14.1
Cost	1172

\[\begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -4.8 \cdot 10^{+58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -3 \cdot 10^{+26}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \mathbf{elif}\;t1 \leq -2.8 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 3.1 \cdot 10^{-164}:\\ \;\;\;\;v \cdot \left(-\frac{\frac{t1}{u}}{u}\right)\\ \mathbf{elif}\;t1 \leq 1.25 \cdot 10^{-53}:\\ \;\;\;\;t1 \cdot \frac{\frac{-v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	13.9
Cost	1105

\[\begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -3.7 \cdot 10^{+62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -2.15 \cdot 10^{+27}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \mathbf{elif}\;t1 \leq -1.4 \cdot 10^{-27} \lor \neg \left(t1 \leq 1.45 \cdot 10^{-55}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{u}{v \cdot \frac{t1}{u}}}\\ \end{array} \]

Alternative 3
Error	13.7
Cost	1105

\[\begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -4.6 \cdot 10^{+58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -2.15 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{v}{u}}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq -4.2 \cdot 10^{-26} \lor \neg \left(t1 \leq 7 \cdot 10^{-56}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{u}{v \cdot \frac{t1}{u}}}\\ \end{array} \]

Alternative 4
Error	13.8
Cost	1105

\[\begin{array}{l} \mathbf{if}\;t1 \leq -2.7 \cdot 10^{+59}:\\ \;\;\;\;v \cdot \frac{1}{u \cdot -2 - t1}\\ \mathbf{elif}\;t1 \leq -6.6 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{v}{u}}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq -7.2 \cdot 10^{-25} \lor \neg \left(t1 \leq 3.6 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{u}{v \cdot \frac{t1}{u}}}\\ \end{array} \]

Alternative 5
Error	13.7
Cost	1104

\[\begin{array}{l} t_1 := u \cdot -2 - t1\\ \mathbf{if}\;t1 \leq -1.2 \cdot 10^{+59}:\\ \;\;\;\;v \cdot \frac{1}{t_1}\\ \mathbf{elif}\;t1 \leq -7.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{v}{u}}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq -1.02 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{\frac{t_1}{v}}\\ \mathbf{elif}\;t1 \leq 1.8 \cdot 10^{-51}:\\ \;\;\;\;\frac{-1}{\frac{u}{v \cdot \frac{t1}{u}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \]

Alternative 6
Error	13.8
Cost	1042

\[\begin{array}{l} \mathbf{if}\;t1 \leq -4.4 \cdot 10^{+58} \lor \neg \left(t1 \leq -1.4 \cdot 10^{+27}\right) \land \left(t1 \leq -6.2 \cdot 10^{-29} \lor \neg \left(t1 \leq 2.15 \cdot 10^{-52}\right)\right):\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \]

Alternative 7
Error	13.9
Cost	1041

\[\begin{array}{l} t_1 := \frac{-v}{u + t1}\\ \mathbf{if}\;t1 \leq -4.4 \cdot 10^{+58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -1.05 \cdot 10^{+27}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \mathbf{elif}\;t1 \leq -6.5 \cdot 10^{-24} \lor \neg \left(t1 \leq 2.4 \cdot 10^{-52}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \]

Alternative 8
Error	20.6
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+162} \lor \neg \left(u \leq 8.2 \cdot 10^{+141}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \end{array} \]

Alternative 9
Error	20.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;u \leq -5.6 \cdot 10^{+58}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{elif}\;u \leq 8.5 \cdot 10^{+141}:\\ \;\;\;\;\frac{-v}{u + t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \end{array} \]

Alternative 10
Error	3.3
Cost	704

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

Alternative 11
Error	1.4
Cost	704

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

Alternative 12
Error	27.7
Cost	521

\[\begin{array}{l} \mathbf{if}\;u \leq -2.35 \cdot 10^{+115} \lor \neg \left(u \leq 7.7 \cdot 10^{+204}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 13
Error	24.8
Cost	384

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

Alternative 14
Error	30.4
Cost	256

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

Reproduce

herbie shell --seed 2023031 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))