Rosa's DopplerBench

Average Error: 17.9 → 1.9

Time: 10.2s

Precision: binary64

Cost: 969

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t1 \leq -3 \cdot 10^{-265} \lor \neg \left(t1 \leq -1.9 \cdot 10^{-298}\right):\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \left(-v\right)}{u \cdot u}\\ \end{array} \]

FPCore

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

↓

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -3e-265) (not (<= t1 -1.9e-298)))
   (/ (/ v (+ t1 u)) (- -1.0 (/ u t1)))
   (/ (* t1 (- v)) (* u 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 tmp;
	if ((t1 <= -3e-265) || !(t1 <= -1.9e-298)) {
		tmp = (v / (t1 + u)) / (-1.0 - (u / t1));
	} else {
		tmp = (t1 * -v) / (u * 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) :: tmp
    if ((t1 <= (-3d-265)) .or. (.not. (t1 <= (-1.9d-298)))) then
        tmp = (v / (t1 + u)) / ((-1.0d0) - (u / t1))
    else
        tmp = (t1 * -v) / (u * 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 tmp;
	if ((t1 <= -3e-265) || !(t1 <= -1.9e-298)) {
		tmp = (v / (t1 + u)) / (-1.0 - (u / t1));
	} else {
		tmp = (t1 * -v) / (u * u);
	}
	return tmp;
}

Python

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

↓

def code(u, v, t1):
	tmp = 0
	if (t1 <= -3e-265) or not (t1 <= -1.9e-298):
		tmp = (v / (t1 + u)) / (-1.0 - (u / t1))
	else:
		tmp = (t1 * -v) / (u * 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)
	tmp = 0.0
	if ((t1 <= -3e-265) || !(t1 <= -1.9e-298))
		tmp = Float64(Float64(v / Float64(t1 + u)) / Float64(-1.0 - Float64(u / t1)));
	else
		tmp = Float64(Float64(t1 * Float64(-v)) / Float64(u * 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)
	tmp = 0.0;
	if ((t1 <= -3e-265) || ~((t1 <= -1.9e-298)))
		tmp = (v / (t1 + u)) / (-1.0 - (u / t1));
	else
		tmp = (t1 * -v) / (u * 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_] := If[Or[LessEqual[t1, -3e-265], N[Not[LessEqual[t1, -1.9e-298]], $MachinePrecision]], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t1 * (-v)), $MachinePrecision] / N[(u * u), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.9999999999999998e-265 or -1.9e-298 < t1

   1. Initial program 17.8

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

   2. Simplified 1.4

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

      [Start] 17.8	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      *-commutative [=>] 17.8	\[ \frac{\color{blue}{v \cdot \left(-t1\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      associate-/l* [=>] 15.6	\[ \color{blue}{\frac{v}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}} \]
      associate-*r/ [<=] 3.6	\[ \frac{v}{\color{blue}{\left(t1 + u\right) \cdot \frac{t1 + u}{-t1}}} \]
      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 [<=] 8.7	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{0 \cdot \frac{t1 + u}{t1}} - 1}} \]
      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 [<=] 15.4	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1 \cdot \left(t1 + u\right)}{-1 \cdot t1}}}{t1}} \]
      neg-mul-1 [<=] 15.4	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 \cdot \left(t1 + u\right)}{\color{blue}{-t1}}}{t1}} \]
      associate-/l* [=>] 1.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1}{\frac{-t1}{t1 + u}}}}{t1}} \]

   if -2.9999999999999998e-265 < t1 < -1.9e-298

   1. Initial program 19.4

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

   2. Taylor expanded in t1 around 0 19.4

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

   3. Simplified 19.4

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

      [Start] 19.4	\[ \frac{\left(-t1\right) \cdot v}{{u}^{2}} \]
      unpow2 [=>] 19.4	\[ \frac{\left(-t1\right) \cdot v}{\color{blue}{u \cdot u}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -3 \cdot 10^{-265} \lor \neg \left(t1 \leq -1.9 \cdot 10^{-298}\right):\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \left(-v\right)}{u \cdot u}\\ \end{array} \]

Alternatives

Alternative 1
Error	16.1
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -0.0023 \lor \neg \left(t1 \leq 1.8 \cdot 10^{-61}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \end{array} \]

Alternative 2
Error	14.6
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -0.0035 \lor \neg \left(t1 \leq 1.3 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \end{array} \]

Alternative 3
Error	14.5
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -0.0017 \lor \neg \left(t1 \leq 1.9 \cdot 10^{-60}\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \end{array} \]

Alternative 4
Error	13.9
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -0.0019 \lor \neg \left(t1 \leq 4.4 \cdot 10^{-60}\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{u}}{u}\\ \end{array} \]

Alternative 5
Error	13.7
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -0.00092 \lor \neg \left(t1 \leq 2 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-t1}{\frac{u}{v}}}{u}\\ \end{array} \]

Alternative 6
Error	1.5
Cost	768

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

Alternative 7
Error	21.8
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -2.65 \cdot 10^{+182} \lor \neg \left(u \leq 6 \cdot 10^{+96}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 8
Error	21.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;u \leq -2.65 \cdot 10^{+182}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{elif}\;u \leq 3.2 \cdot 10^{+95}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \end{array} \]

Alternative 9
Error	3.6
Cost	704

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

Alternative 10
Error	3.6
Cost	704

\[\frac{v}{u \cdot \left(-2 - \frac{u}{t1}\right) - t1} \]

Alternative 11
Error	28.8
Cost	521

\[\begin{array}{l} \mathbf{if}\;u \leq -3 \cdot 10^{+160} \lor \neg \left(u \leq 1.12 \cdot 10^{+94}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 12
Error	26.2
Cost	384

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

Alternative 13
Error	32.0
Cost	256

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

Reproduce

herbie shell --seed 2023073 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))