Rosa's DopplerBench

Average Error: 18.4 → 6.7

Time: 21.7s

Precision: binary64

Cost: 7364

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t1 \leq -5.5 \cdot 10^{+170}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 + \left(\left(-t1\right) - \frac{{u}^{2}}{t1}\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (u v t1)
 :precision binary64
 (if (<= t1 -5.5e+170)
   (- (/ v t1))
   (/ v (+ (* u -2.0) (- (- t1) (/ (pow u 2.0) 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 tmp;
	if (t1 <= -5.5e+170) {
		tmp = -(v / t1);
	} else {
		tmp = v / ((u * -2.0) + (-t1 - (pow(u, 2.0) / 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) :: tmp
    if (t1 <= (-5.5d+170)) then
        tmp = -(v / t1)
    else
        tmp = v / ((u * (-2.0d0)) + (-t1 - ((u ** 2.0d0) / 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 tmp;
	if (t1 <= -5.5e+170) {
		tmp = -(v / t1);
	} else {
		tmp = v / ((u * -2.0) + (-t1 - (Math.pow(u, 2.0) / t1)));
	}
	return tmp;
}

Python

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

↓

def code(u, v, t1):
	tmp = 0
	if t1 <= -5.5e+170:
		tmp = -(v / t1)
	else:
		tmp = v / ((u * -2.0) + (-t1 - (math.pow(u, 2.0) / 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)
	tmp = 0.0
	if (t1 <= -5.5e+170)
		tmp = Float64(-Float64(v / t1));
	else
		tmp = Float64(v / Float64(Float64(u * -2.0) + Float64(Float64(-t1) - Float64((u ^ 2.0) / 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)
	tmp = 0.0;
	if (t1 <= -5.5e+170)
		tmp = -(v / t1);
	else
		tmp = v / ((u * -2.0) + (-t1 - ((u ^ 2.0) / 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_] := If[LessEqual[t1, -5.5e+170], (-N[(v / t1), $MachinePrecision]), N[(v / N[(N[(u * -2.0), $MachinePrecision] + N[((-t1) - N[(N[Power[u, 2.0], $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -5.4999999999999999e170

   1. Initial program 39.9

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

   2. Taylor expanded in t1 around inf 5.5

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

   3. Simplified 5.5

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

      [Start] 5.5	\[ -1 \cdot \frac{v}{t1} \]
      rational.json-simplify-2 [=>] 5.5	\[ \color{blue}{\frac{v}{t1} \cdot -1} \]
      rational.json-simplify-9 [=>] 5.5	\[ \color{blue}{-\frac{v}{t1}} \]

   if -5.4999999999999999e170 < t1

   1. Initial program 15.6

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

   2. Applied egg-rr 48.9

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

   3. Applied egg-rr 48.6

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

   4. Taylor expanded in t1 around 0 48.5

      \[\leadsto \frac{1}{\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot \left(\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot \color{blue}{\left(-1 \cdot \frac{{u}^{2}}{t1 \cdot v} + \left(-2 \cdot \frac{u}{v} + -1 \cdot \frac{t1}{v}\right)\right)}\right)} \cdot \left(\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot \frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right) \]

   5. Simplified 48.5

      \[\leadsto \frac{1}{\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot \left(\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot \color{blue}{\left(-2 \cdot \frac{u}{v} + -1 \cdot \left(\frac{{u}^{2}}{t1 \cdot v} + \frac{t1}{v}\right)\right)}\right)} \cdot \left(\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot \frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right) \]
      Proof

      [Start] 48.5	\[ \frac{1}{\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot \left(\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot \left(-1 \cdot \frac{{u}^{2}}{t1 \cdot v} + \left(-2 \cdot \frac{u}{v} + -1 \cdot \frac{t1}{v}\right)\right)\right)} \cdot \left(\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot \frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right) \]
      rational.json-simplify-41 [=>] 48.5	\[ \frac{1}{\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot \left(\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot \color{blue}{\left(-2 \cdot \frac{u}{v} + \left(-1 \cdot \frac{t1}{v} + -1 \cdot \frac{{u}^{2}}{t1 \cdot v}\right)\right)}\right)} \cdot \left(\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot \frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right) \]
      rational.json-simplify-2 [=>] 48.5	\[ \frac{1}{\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot \left(\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot \left(-2 \cdot \frac{u}{v} + \left(-1 \cdot \frac{t1}{v} + \color{blue}{\frac{{u}^{2}}{t1 \cdot v} \cdot -1}\right)\right)\right)} \cdot \left(\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot \frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right) \]
      rational.json-simplify-47 [=>] 48.5	\[ \frac{1}{\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot \left(\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot \left(-2 \cdot \frac{u}{v} + \color{blue}{-1 \cdot \left(\frac{{u}^{2}}{t1 \cdot v} + \frac{t1}{v}\right)}\right)\right)} \cdot \left(\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot \frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\right) \]

   6. Taylor expanded in v around 0 6.9

      \[\leadsto \color{blue}{\frac{v}{-2 \cdot u + -1 \cdot \left(t1 + \frac{{u}^{2}}{t1}\right)}} \]

   7. Simplified 6.9

      \[\leadsto \color{blue}{\frac{v}{u \cdot -2 + \left(\left(-t1\right) - \frac{{u}^{2}}{t1}\right)}} \]
      Proof

      [Start] 6.9	\[ \frac{v}{-2 \cdot u + -1 \cdot \left(t1 + \frac{{u}^{2}}{t1}\right)} \]
      rational.json-simplify-2 [=>] 6.9	\[ \frac{v}{\color{blue}{u \cdot -2} + -1 \cdot \left(t1 + \frac{{u}^{2}}{t1}\right)} \]
      rational.json-simplify-2 [=>] 6.9	\[ \frac{v}{u \cdot -2 + \color{blue}{\left(t1 + \frac{{u}^{2}}{t1}\right) \cdot -1}} \]
      rational.json-simplify-9 [=>] 6.9	\[ \frac{v}{u \cdot -2 + \color{blue}{\left(-\left(t1 + \frac{{u}^{2}}{t1}\right)\right)}} \]
      rational.json-simplify-12 [=>] 6.9	\[ \frac{v}{u \cdot -2 + \color{blue}{\left(0 - \left(t1 + \frac{{u}^{2}}{t1}\right)\right)}} \]
      rational.json-simplify-46 [=>] 6.9	\[ \frac{v}{u \cdot -2 + \color{blue}{\left(\left(0 - t1\right) - \frac{{u}^{2}}{t1}\right)}} \]
      rational.json-simplify-12 [<=] 6.9	\[ \frac{v}{u \cdot -2 + \left(\color{blue}{\left(-t1\right)} - \frac{{u}^{2}}{t1}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -5.5 \cdot 10^{+170}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 + \left(\left(-t1\right) - \frac{{u}^{2}}{t1}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.6
Cost	1032

\[\begin{array}{l} t_1 := -\frac{v}{t1}\\ \mathbf{if}\;t1 \leq -6.8 \cdot 10^{+140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 3 \cdot 10^{+92}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	28.3
Cost	968

\[\begin{array}{l} t_1 := \left(0 - \left(-1 - v \cdot \frac{-1}{t1}\right)\right) - 1\\ \mathbf{if}\;u \leq -5.5 \cdot 10^{+123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 8.5 \cdot 10^{+56}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	31.0
Cost	256

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

Reproduce

herbie shell --seed 2023077 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))