Rosa's DopplerBench

Average Accuracy: 71.9% → 96.8%

Time: 14.7s

Precision: binary64

Cost: 969

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t1 \leq -2.4 \cdot 10^{-222} \lor \neg \left(t1 \leq 1.22 \cdot 10^{-238}\right):\\ \;\;\;\;\frac{\frac{v}{-1 - \frac{u}{t1}}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \end{array} \]

FPCore

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

↓

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -2.4e-222) (not (<= t1 1.22e-238)))
   (/ (/ v (- -1.0 (/ u t1))) (+ t1 u))
   (/ (- t1) (* u (/ u v)))))

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 <= -2.4e-222) || !(t1 <= 1.22e-238)) {
		tmp = (v / (-1.0 - (u / t1))) / (t1 + u);
	} else {
		tmp = -t1 / (u * (u / v));
	}
	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 <= (-2.4d-222)) .or. (.not. (t1 <= 1.22d-238))) then
        tmp = (v / ((-1.0d0) - (u / t1))) / (t1 + u)
    else
        tmp = -t1 / (u * (u / v))
    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 <= -2.4e-222) || !(t1 <= 1.22e-238)) {
		tmp = (v / (-1.0 - (u / t1))) / (t1 + u);
	} else {
		tmp = -t1 / (u * (u / v));
	}
	return tmp;
}

Python

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

↓

def code(u, v, t1):
	tmp = 0
	if (t1 <= -2.4e-222) or not (t1 <= 1.22e-238):
		tmp = (v / (-1.0 - (u / t1))) / (t1 + u)
	else:
		tmp = -t1 / (u * (u / v))
	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 <= -2.4e-222) || !(t1 <= 1.22e-238))
		tmp = Float64(Float64(v / Float64(-1.0 - Float64(u / t1))) / Float64(t1 + u));
	else
		tmp = Float64(Float64(-t1) / Float64(u * Float64(u / v)));
	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 <= -2.4e-222) || ~((t1 <= 1.22e-238)))
		tmp = (v / (-1.0 - (u / t1))) / (t1 + u);
	else
		tmp = -t1 / (u * (u / v));
	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, -2.4e-222], N[Not[LessEqual[t1, 1.22e-238]], $MachinePrecision]], N[(N[(v / N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t1 + u), $MachinePrecision]), $MachinePrecision], N[((-t1) / N[(u * N[(u / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -2.39999999999999993e-222 or 1.22e-238 < t1

   1. Initial program 71.5%

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

   2. Simplified 98.8%

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

      [Start] 71.5	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      *-commutative [=>] 71.5	\[ \frac{\color{blue}{v \cdot \left(-t1\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      associate-/l* [=>] 75.5	\[ \color{blue}{\frac{v}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}} \]
      associate-*r/ [<=] 95.2	\[ \frac{v}{\color{blue}{\left(t1 + u\right) \cdot \frac{t1 + u}{-t1}}} \]
      associate-/r* [=>] 98.8	\[ \color{blue}{\frac{\frac{v}{t1 + u}}{\frac{t1 + u}{-t1}}} \]
      neg-mul-1 [=>] 98.8	\[ \frac{\frac{v}{t1 + u}}{\frac{t1 + u}{\color{blue}{-1 \cdot t1}}} \]
      associate-/l/ [<=] 98.8	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{\frac{t1 + u}{t1}}{-1}}} \]
      metadata-eval [<=] 98.8	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{0 - 1}}} \]
      mul0-lft [<=] 91.4	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{0 \cdot \frac{t1 + u}{t1}} - 1}} \]
      associate-*r/ [=>] 98.8	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{0 \cdot \left(t1 + u\right)}{t1}} - 1}} \]
      mul0-lft [=>] 98.8	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{0}}{t1} - 1}} \]
      *-inverses [<=] 98.8	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{0}{t1} - \color{blue}{\frac{t1}{t1}}}} \]
      div-sub [<=] 98.8	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{0 - t1}{t1}}}} \]
      neg-sub0 [<=] 98.8	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{-t1}}{t1}}} \]
      neg-mul-1 [=>] 98.8	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{-1 \cdot t1}}{t1}}} \]
      *-commutative [=>] 98.8	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{t1 \cdot -1}}{t1}}} \]
      associate-/l* [=>] 98.8	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{t1}{\frac{t1}{-1}}}}} \]
      associate-/l* [<=] 98.8	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{\frac{t1 + u}{t1} \cdot \frac{t1}{-1}}{t1}}} \]
      *-commutative [=>] 98.8	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1}{-1} \cdot \frac{t1 + u}{t1}}}{t1}} \]
      times-frac [<=] 77.3	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1 \cdot \left(t1 + u\right)}{-1 \cdot t1}}}{t1}} \]
      neg-mul-1 [<=] 77.3	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 \cdot \left(t1 + u\right)}{\color{blue}{-t1}}}{t1}} \]
      associate-/l* [=>] 98.7	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1}{\frac{-t1}{t1 + u}}}}{t1}} \]

   3. Applied egg-rr 98.8%

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

      [Start] 98.8	\[ \frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}} \]
      div-inv [=>] 98.6	\[ \frac{\color{blue}{v \cdot \frac{1}{t1 + u}}}{-1 - \frac{u}{t1}} \]
      associate-/l* [=>] 95.0	\[ \color{blue}{\frac{v}{\frac{-1 - \frac{u}{t1}}{\frac{1}{t1 + u}}}} \]
      associate-/r/ [=>] 98.8	\[ \color{blue}{\frac{v}{-1 - \frac{u}{t1}} \cdot \frac{1}{t1 + u}} \]

   4. Applied egg-rr 99.0%

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

      [Start] 98.8	\[ \frac{v}{-1 - \frac{u}{t1}} \cdot \frac{1}{t1 + u} \]
      un-div-inv [=>] 99.0	\[ \color{blue}{\frac{\frac{v}{-1 - \frac{u}{t1}}}{t1 + u}} \]
      +-commutative [=>] 99.0	\[ \frac{\frac{v}{-1 - \frac{u}{t1}}}{\color{blue}{u + t1}} \]

   if -2.39999999999999993e-222 < t1 < 1.22e-238

   1. Initial program 74.4%

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

   2. Simplified 76.3%

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

      [Start] 74.4	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      associate-*l/ [<=] 76.3	\[ \color{blue}{\frac{-t1}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \cdot v} \]

   3. Taylor expanded in t1 around 0 74.4%

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

   4. Simplified 81.3%

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

      [Start] 74.4	\[ -1 \cdot \frac{t1 \cdot v}{{u}^{2}} \]
      associate-/l* [=>] 74.9	\[ -1 \cdot \color{blue}{\frac{t1}{\frac{{u}^{2}}{v}}} \]
      associate-*r/ [=>] 74.9	\[ \color{blue}{\frac{-1 \cdot t1}{\frac{{u}^{2}}{v}}} \]
      mul-1-neg [=>] 74.9	\[ \frac{\color{blue}{-t1}}{\frac{{u}^{2}}{v}} \]
      unpow2 [=>] 74.9	\[ \frac{-t1}{\frac{\color{blue}{u \cdot u}}{v}} \]
      associate-/l* [=>] 81.3	\[ \frac{-t1}{\color{blue}{\frac{u}{\frac{v}{u}}}} \]

   5. Applied egg-rr 81.3%

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

      [Start] 81.3	\[ \frac{-t1}{\frac{u}{\frac{v}{u}}} \]
      associate-/r/ [=>] 81.3	\[ \frac{-t1}{\color{blue}{\frac{u}{v} \cdot u}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.4 \cdot 10^{-222} \lor \neg \left(t1 \leq 1.22 \cdot 10^{-238}\right):\\ \;\;\;\;\frac{\frac{v}{-1 - \frac{u}{t1}}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	65.2%
Cost	978

\[\begin{array}{l} \mathbf{if}\;t1 \leq -2.1 \cdot 10^{-11} \lor \neg \left(t1 \leq -2.8 \cdot 10^{-67}\right) \land \left(t1 \leq -5.4 \cdot 10^{-120} \lor \neg \left(t1 \leq 4.2 \cdot 10^{-196}\right)\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \end{array} \]

Alternative 2
Accuracy	65.4%
Cost	977

\[\begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -5.6 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -2.5 \cdot 10^{-58}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{elif}\;t1 \leq -2.6 \cdot 10^{-118} \lor \neg \left(t1 \leq 3.2 \cdot 10^{-195}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \]

Alternative 3
Accuracy	65.6%
Cost	977

\[\begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -6.3 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -3.9 \cdot 10^{-59}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{elif}\;t1 \leq -9.8 \cdot 10^{-119} \lor \neg \left(t1 \leq 5.2 \cdot 10^{-195}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \]

Alternative 4
Accuracy	94.1%
Cost	969

\[\begin{array}{l} \mathbf{if}\;v \leq 1.85 \cdot 10^{+109} \lor \neg \left(v \leq 1.3 \cdot 10^{+141}\right):\\ \;\;\;\;\frac{v}{\left(-1 - \frac{u}{t1}\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{t1 + u}}{u}\\ \end{array} \]

Alternative 5
Accuracy	95.7%
Cost	969

\[\begin{array}{l} \mathbf{if}\;t1 \leq -7 \cdot 10^{-168} \lor \neg \left(t1 \leq 7 \cdot 10^{-231}\right):\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{\frac{u}{\frac{v}{u}}}\\ \end{array} \]

Alternative 6
Accuracy	79.4%
Cost	905

\[\begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{-9} \lor \neg \left(t1 \leq 3.3 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{-v}{t1 + u}}{u}\\ \end{array} \]

Alternative 7
Accuracy	79.4%
Cost	841

\[\begin{array}{l} \mathbf{if}\;t1 \leq -5.8 \cdot 10^{-8} \lor \neg \left(t1 \leq 9.5 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{u} \cdot \left(t1 \cdot \frac{v}{u}\right)\\ \end{array} \]

Alternative 8
Accuracy	76.3%
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -1.12 \cdot 10^{-10} \lor \neg \left(t1 \leq 1.4 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{\frac{u \cdot u}{v}}\\ \end{array} \]

Alternative 9
Accuracy	79.5%
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -6.8 \cdot 10^{-10} \lor \neg \left(t1 \leq 5.6 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u} \cdot \frac{t1}{u}\\ \end{array} \]

Alternative 10
Accuracy	78.2%
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -9.5 \cdot 10^{-8} \lor \neg \left(t1 \leq 8.9 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \end{array} \]

Alternative 11
Accuracy	57.3%
Cost	521

\[\begin{array}{l} \mathbf{if}\;u \leq -6.5 \cdot 10^{+179} \lor \neg \left(u \leq 9.5 \cdot 10^{+191}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 12
Accuracy	61.0%
Cost	384

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

Alternative 13
Accuracy	52.9%
Cost	256

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

Reproduce

herbie shell --seed 2023137 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))