Rosa's DopplerBench

Average Accuracy: 69.3% → 94.2%

Time: 11.9s

Precision: binary64

Cost: 1032

Math

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

↓

\[\begin{array}{l} t_1 := \frac{v}{t1 + u}\\ \mathbf{if}\;t1 \leq -2.2 \cdot 10^{-251}:\\ \;\;\;\;\frac{t_1}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 2.25 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{t1 \cdot v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{t1 + u} \cdot t_1\\ \end{array} \]

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 -2.2e-251)
     (/ t_1 (- -1.0 (/ u t1)))
     (if (<= t1 2.25e-269)
       (/ (/ (* t1 v) u) (- u))
       (* (/ (- t1) (+ t1 u)) t_1)))))

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 <= -2.2e-251) {
		tmp = t_1 / (-1.0 - (u / t1));
	} else if (t1 <= 2.25e-269) {
		tmp = ((t1 * v) / u) / -u;
	} else {
		tmp = (-t1 / (t1 + u)) * t_1;
	}
	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 <= (-2.2d-251)) then
        tmp = t_1 / ((-1.0d0) - (u / t1))
    else if (t1 <= 2.25d-269) then
        tmp = ((t1 * v) / u) / -u
    else
        tmp = (-t1 / (t1 + u)) * t_1
    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 <= -2.2e-251) {
		tmp = t_1 / (-1.0 - (u / t1));
	} else if (t1 <= 2.25e-269) {
		tmp = ((t1 * v) / u) / -u;
	} else {
		tmp = (-t1 / (t1 + u)) * t_1;
	}
	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 <= -2.2e-251:
		tmp = t_1 / (-1.0 - (u / t1))
	elif t1 <= 2.25e-269:
		tmp = ((t1 * v) / u) / -u
	else:
		tmp = (-t1 / (t1 + u)) * t_1
	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 <= -2.2e-251)
		tmp = Float64(t_1 / Float64(-1.0 - Float64(u / t1)));
	elseif (t1 <= 2.25e-269)
		tmp = Float64(Float64(Float64(t1 * v) / u) / Float64(-u));
	else
		tmp = Float64(Float64(Float64(-t1) / Float64(t1 + u)) * t_1);
	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 <= -2.2e-251)
		tmp = t_1 / (-1.0 - (u / t1));
	elseif (t1 <= 2.25e-269)
		tmp = ((t1 * v) / u) / -u;
	else
		tmp = (-t1 / (t1 + u)) * t_1;
	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, -2.2e-251], N[(t$95$1 / N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t1, 2.25e-269], N[(N[(N[(t1 * v), $MachinePrecision] / u), $MachinePrecision] / (-u)), $MachinePrecision], N[(N[((-t1) / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t1 < -2.2e-251

   1. Initial program 63.9%

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

   2. Simplified 93.1%

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

      [Start] 63.9	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      *-commutative [=>] 63.9	\[ \frac{\color{blue}{v \cdot \left(-t1\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      times-frac [=>] 93.0	\[ \color{blue}{\frac{v}{t1 + u} \cdot \frac{-t1}{t1 + u}} \]
      neg-mul-1 [=>] 93.0	\[ \frac{v}{t1 + u} \cdot \frac{\color{blue}{-1 \cdot t1}}{t1 + u} \]
      associate-/l* [=>] 92.9	\[ \frac{v}{t1 + u} \cdot \color{blue}{\frac{-1}{\frac{t1 + u}{t1}}} \]
      associate-*r/ [=>] 93.1	\[ \color{blue}{\frac{\frac{v}{t1 + u} \cdot -1}{\frac{t1 + u}{t1}}} \]
      associate-/l* [=>] 93.1	\[ \color{blue}{\frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{-1}}} \]
      associate-/l/ [=>] 93.1	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{t1 + u}{-1 \cdot t1}}} \]
      neg-mul-1 [<=] 93.1	\[ \frac{\frac{v}{t1 + u}}{\frac{t1 + u}{\color{blue}{-t1}}} \]
      *-lft-identity [<=] 93.1	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{1 \cdot \frac{t1 + u}{-t1}}} \]
      metadata-eval [<=] 93.1	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{-1}{-1}} \cdot \frac{t1 + u}{-t1}} \]
      times-frac [<=] 93.1	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{-1 \cdot \left(t1 + u\right)}{-1 \cdot \left(-t1\right)}}} \]
      neg-mul-1 [<=] 93.1	\[ \frac{\frac{v}{t1 + u}}{\frac{-1 \cdot \left(t1 + u\right)}{\color{blue}{-\left(-t1\right)}}} \]
      remove-double-neg [=>] 93.1	\[ \frac{\frac{v}{t1 + u}}{\frac{-1 \cdot \left(t1 + u\right)}{\color{blue}{t1}}} \]
      neg-mul-1 [<=] 93.1	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{-\left(t1 + u\right)}}{t1}} \]
      sub0-neg [<=] 93.1	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - \left(t1 + u\right)}}{t1}} \]
      associate--r+ [=>] 93.1	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(0 - t1\right) - u}}{t1}} \]
      neg-sub0 [<=] 93.1	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(-t1\right)} - u}{t1}} \]
      div-sub [=>] 93.1	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{-t1}{t1} - \frac{u}{t1}}} \]
      distribute-frac-neg [=>] 93.1	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\left(-\frac{t1}{t1}\right)} - \frac{u}{t1}} \]
      *-inverses [=>] 93.1	\[ \frac{\frac{v}{t1 + u}}{\left(-\color{blue}{1}\right) - \frac{u}{t1}} \]
      metadata-eval [=>] 93.1	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{-1} - \frac{u}{t1}} \]

   if -2.2e-251 < t1 < 2.2500000000000001e-269

   1. Initial program 85.8%

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

   2. Simplified 85.6%

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

      [Start] 85.8	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      associate-/l* [=>] 85.6	\[ \color{blue}{\frac{-t1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}}} \]
      neg-mul-1 [=>] 85.6	\[ \frac{\color{blue}{-1 \cdot t1}}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}} \]
      *-commutative [=>] 85.6	\[ \frac{\color{blue}{t1 \cdot -1}}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}} \]
      associate-*r/ [<=] 85.6	\[ \color{blue}{t1 \cdot \frac{-1}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{v}}} \]
      associate-/l* [<=] 85.7	\[ t1 \cdot \color{blue}{\frac{-1 \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]
      neg-mul-1 [<=] 85.7	\[ t1 \cdot \frac{\color{blue}{-v}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      associate-/r* [=>] 85.6	\[ t1 \cdot \color{blue}{\frac{\frac{-v}{t1 + u}}{t1 + u}} \]

   3. Taylor expanded in t1 around 0 85.7%

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

   4. Simplified 85.6%

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

      [Start] 85.7	\[ t1 \cdot \left(-1 \cdot \frac{v}{{u}^{2}}\right) \]
      associate-*r/ [=>] 85.7	\[ t1 \cdot \color{blue}{\frac{-1 \cdot v}{{u}^{2}}} \]
      unpow2 [=>] 85.7	\[ t1 \cdot \frac{-1 \cdot v}{\color{blue}{u \cdot u}} \]
      associate-/r* [=>] 85.6	\[ t1 \cdot \color{blue}{\frac{\frac{-1 \cdot v}{u}}{u}} \]
      neg-mul-1 [<=] 85.6	\[ t1 \cdot \frac{\frac{\color{blue}{-v}}{u}}{u} \]

   5. Applied egg-rr 89.4%

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

      [Start] 85.6	\[ t1 \cdot \frac{\frac{-v}{u}}{u} \]
      *-commutative [=>] 85.6	\[ \color{blue}{\frac{\frac{-v}{u}}{u} \cdot t1} \]
      frac-2neg [=>] 85.6	\[ \color{blue}{\frac{-\frac{-v}{u}}{-u}} \cdot t1 \]
      associate-*l/ [=>] 85.7	\[ \color{blue}{\frac{\left(-\frac{-v}{u}\right) \cdot t1}{-u}} \]
      distribute-neg-frac [=>] 85.7	\[ \frac{\color{blue}{\frac{-\left(-v\right)}{u}} \cdot t1}{-u} \]
      remove-double-neg [=>] 85.7	\[ \frac{\frac{\color{blue}{v}}{u} \cdot t1}{-u} \]
      add-sqr-sqrt [=>] 49.8	\[ \frac{\frac{\color{blue}{\sqrt{v} \cdot \sqrt{v}}}{u} \cdot t1}{-u} \]
      sqrt-unprod [=>] 57.1	\[ \frac{\frac{\color{blue}{\sqrt{v \cdot v}}}{u} \cdot t1}{-u} \]
      sqr-neg [<=] 57.1	\[ \frac{\frac{\sqrt{\color{blue}{\left(-v\right) \cdot \left(-v\right)}}}{u} \cdot t1}{-u} \]
      sqrt-unprod [<=] 32.7	\[ \frac{\frac{\color{blue}{\sqrt{-v} \cdot \sqrt{-v}}}{u} \cdot t1}{-u} \]
      add-sqr-sqrt [<=] 59.2	\[ \frac{\frac{\color{blue}{-v}}{u} \cdot t1}{-u} \]
      *-commutative [<=] 59.2	\[ \frac{\color{blue}{t1 \cdot \frac{-v}{u}}}{-u} \]
      associate-*r/ [=>] 59.2	\[ \frac{\color{blue}{\frac{t1 \cdot \left(-v\right)}{u}}}{-u} \]
      add-sqr-sqrt [=>] 32.7	\[ \frac{\frac{t1 \cdot \color{blue}{\left(\sqrt{-v} \cdot \sqrt{-v}\right)}}{u}}{-u} \]
      sqrt-unprod [=>] 53.8	\[ \frac{\frac{t1 \cdot \color{blue}{\sqrt{\left(-v\right) \cdot \left(-v\right)}}}{u}}{-u} \]
      sqr-neg [=>] 53.8	\[ \frac{\frac{t1 \cdot \sqrt{\color{blue}{v \cdot v}}}{u}}{-u} \]
      sqrt-unprod [<=] 50.1	\[ \frac{\frac{t1 \cdot \color{blue}{\left(\sqrt{v} \cdot \sqrt{v}\right)}}{u}}{-u} \]
      add-sqr-sqrt [<=] 89.4	\[ \frac{\frac{t1 \cdot \color{blue}{v}}{u}}{-u} \]

   if 2.2500000000000001e-269 < t1

   1. Initial program 72.7%

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

   2. Simplified 99.1%

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

      [Start] 72.7	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      times-frac [=>] 99.1	\[ \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -2.2 \cdot 10^{-251}:\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 2.25 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{t1 \cdot v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	94.1%
Cost	969

\[\begin{array}{l} \mathbf{if}\;t1 \leq -9.2 \cdot 10^{-251} \lor \neg \left(t1 \leq 3.15 \cdot 10^{-270}\right):\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1 \cdot v}{u}}{-u}\\ \end{array} \]

Alternative 2
Accuracy	74.5%
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -9 \cdot 10^{+31} \lor \neg \left(t1 \leq 0.2\right):\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{u}}{u}\\ \end{array} \]

Alternative 3
Accuracy	74.6%
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -1.1 \cdot 10^{+36} \lor \neg \left(t1 \leq 0.076\right):\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{\frac{u}{\frac{v}{u}}}\\ \end{array} \]

Alternative 4
Accuracy	73.3%
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -6.5 \cdot 10^{+120} \lor \neg \left(t1 \leq 0.9\right):\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \end{array} \]

Alternative 5
Accuracy	65.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -2.2 \cdot 10^{+124} \lor \neg \left(u \leq 3.8 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{t1}{\frac{u \cdot u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 6
Accuracy	63.4%
Cost	712

\[\begin{array}{l} \mathbf{if}\;t1 \leq -8.5 \cdot 10^{-183}:\\ \;\;\;\;\frac{v}{u - t1}\\ \mathbf{elif}\;t1 \leq 5.1 \cdot 10^{-172}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 7
Accuracy	55.4%
Cost	585

\[\begin{array}{l} \mathbf{if}\;u \leq -9 \cdot 10^{+140} \lor \neg \left(u \leq 5.8 \cdot 10^{+162}\right):\\ \;\;\;\;\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 8
Accuracy	55.1%
Cost	520

\[\begin{array}{l} \mathbf{if}\;u \leq -1.15 \cdot 10^{+145}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 3.1 \cdot 10^{+186}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 9
Accuracy	21.4%
Cost	456

\[\begin{array}{l} \mathbf{if}\;t1 \leq -6.7 \cdot 10^{+168}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 5.8 \cdot 10^{+119}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \]

Alternative 10
Accuracy	58.7%
Cost	320

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

Alternative 11
Accuracy	14.4%
Cost	192

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

Reproduce

herbie shell --seed 2023157 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))