Rosa's DopplerBench

Average Error: 27.98% → 2.19%

Time: 11.5s

Precision: binary64

Cost: 2440

Math

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

↓

\[\begin{array}{l} t_1 := \frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{v}{-1 - \frac{u}{t1}} \cdot \frac{1}{t1 + u}\\ \mathbf{elif}\;t_1 \leq 10^{+187}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}\\ \end{array} \]

FPCore

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

↓

(FPCore (u v t1)
 :precision binary64
 (let* ((t_1 (/ (* t1 (- v)) (* (+ t1 u) (+ t1 u)))))
   (if (<= t_1 0.0)
     (* (/ v (- -1.0 (/ u t1))) (/ 1.0 (+ t1 u)))
     (if (<= t_1 1e+187) t_1 (* (/ (- t1) (+ t1 u)) (/ v (+ t1 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 t_1 = (t1 * -v) / ((t1 + u) * (t1 + u));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (v / (-1.0 - (u / t1))) * (1.0 / (t1 + u));
	} else if (t_1 <= 1e+187) {
		tmp = t_1;
	} else {
		tmp = (-t1 / (t1 + u)) * (v / (t1 + 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) :: t_1
    real(8) :: tmp
    t_1 = (t1 * -v) / ((t1 + u) * (t1 + u))
    if (t_1 <= 0.0d0) then
        tmp = (v / ((-1.0d0) - (u / t1))) * (1.0d0 / (t1 + u))
    else if (t_1 <= 1d+187) then
        tmp = t_1
    else
        tmp = (-t1 / (t1 + u)) * (v / (t1 + 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 t_1 = (t1 * -v) / ((t1 + u) * (t1 + u));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (v / (-1.0 - (u / t1))) * (1.0 / (t1 + u));
	} else if (t_1 <= 1e+187) {
		tmp = t_1;
	} else {
		tmp = (-t1 / (t1 + u)) * (v / (t1 + u));
	}
	return tmp;
}

Python

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

↓

def code(u, v, t1):
	t_1 = (t1 * -v) / ((t1 + u) * (t1 + u))
	tmp = 0
	if t_1 <= 0.0:
		tmp = (v / (-1.0 - (u / t1))) * (1.0 / (t1 + u))
	elif t_1 <= 1e+187:
		tmp = t_1
	else:
		tmp = (-t1 / (t1 + u)) * (v / (t1 + 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)
	t_1 = Float64(Float64(t1 * Float64(-v)) / Float64(Float64(t1 + u) * Float64(t1 + u)))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(v / Float64(-1.0 - Float64(u / t1))) * Float64(1.0 / Float64(t1 + u)));
	elseif (t_1 <= 1e+187)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(-t1) / Float64(t1 + u)) * Float64(v / Float64(t1 + 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)
	t_1 = (t1 * -v) / ((t1 + u) * (t1 + u));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = (v / (-1.0 - (u / t1))) * (1.0 / (t1 + u));
	elseif (t_1 <= 1e+187)
		tmp = t_1;
	else
		tmp = (-t1 / (t1 + u)) * (v / (t1 + 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_] := Block[{t$95$1 = N[(N[(t1 * (-v)), $MachinePrecision] / N[(N[(t1 + u), $MachinePrecision] * N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(v / N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+187], t$95$1, N[(N[((-t1) / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (neg.f64 t1) v) (*.f64 (+.f64 t1 u) (+.f64 t1 u))) < 0.0

   1. Initial program 19.69

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

   2. Simplified 1.84

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

      [Start] 19.69	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      *-commutative [=>] 19.69	\[ \frac{\color{blue}{v \cdot \left(-t1\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      associate-/l* [=>] 15.99	\[ \color{blue}{\frac{v}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}} \]
      associate-*r/ [<=] 5.26	\[ \frac{v}{\color{blue}{\left(t1 + u\right) \cdot \frac{t1 + u}{-t1}}} \]
      associate-/r* [=>] 1.85	\[ \color{blue}{\frac{\frac{v}{t1 + u}}{\frac{t1 + u}{-t1}}} \]
      neg-mul-1 [=>] 1.85	\[ \frac{\frac{v}{t1 + u}}{\frac{t1 + u}{\color{blue}{-1 \cdot t1}}} \]
      associate-/l/ [<=] 1.85	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{\frac{t1 + u}{t1}}{-1}}} \]
      metadata-eval [<=] 1.85	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{0 - 1}}} \]
      mul0-lft [<=] 19.28	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{0 \cdot \frac{t1 + u}{t1}} - 1}} \]
      associate-*r/ [=>] 1.85	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{0 \cdot \left(t1 + u\right)}{t1}} - 1}} \]
      mul0-lft [=>] 1.85	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{0}}{t1} - 1}} \]
      *-inverses [<=] 1.85	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{0}{t1} - \color{blue}{\frac{t1}{t1}}}} \]
      div-sub [<=] 1.85	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{0 - t1}{t1}}}} \]
      neg-sub0 [<=] 1.85	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{-t1}}{t1}}} \]
      neg-mul-1 [=>] 1.85	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{-1 \cdot t1}}{t1}}} \]
      *-commutative [=>] 1.85	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{t1 \cdot -1}}{t1}}} \]
      associate-/l* [=>] 1.85	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{t1}{\frac{t1}{-1}}}}} \]
      associate-/l* [<=] 1.85	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{\frac{t1 + u}{t1} \cdot \frac{t1}{-1}}{t1}}} \]
      *-commutative [=>] 1.85	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1}{-1} \cdot \frac{t1 + u}{t1}}}{t1}} \]
      times-frac [<=] 14.6	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1 \cdot \left(t1 + u\right)}{-1 \cdot t1}}}{t1}} \]
      neg-mul-1 [<=] 14.6	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 \cdot \left(t1 + u\right)}{\color{blue}{-t1}}}{t1}} \]
      associate-/l* [=>] 1.86	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1}{\frac{-t1}{t1 + u}}}}{t1}} \]

   3. Applied egg-rr 2.32

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

   if 0.0 < (/.f64 (*.f64 (neg.f64 t1) v) (*.f64 (+.f64 t1 u) (+.f64 t1 u))) < 9.99999999999999907e186

   1. Initial program 1.13

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

   if 9.99999999999999907e186 < (/.f64 (*.f64 (neg.f64 t1) v) (*.f64 (+.f64 t1 u) (+.f64 t1 u)))

   1. Initial program 87.57

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

   2. Simplified 2.72

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

      [Start] 87.57	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      times-frac [=>] 2.72	\[ \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 2.19

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \leq 0:\\ \;\;\;\;\frac{v}{-1 - \frac{u}{t1}} \cdot \frac{1}{t1 + u}\\ \mathbf{elif}\;\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \leq 10^{+187}:\\ \;\;\;\;\frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.87%
Cost	2440

\[\begin{array}{l} t_1 := \frac{t1 \cdot \left(-v\right)}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\\ t_2 := \frac{v}{t1 + u}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{t_2}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t_1 \leq 10^{+187}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{t1 + u} \cdot t_2\\ \end{array} \]

Alternative 2
Error	2.93%
Cost	969

\[\begin{array}{l} \mathbf{if}\;t1 \leq -6.5 \cdot 10^{-257} \lor \neg \left(t1 \leq 2 \cdot 10^{-205}\right):\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{u} \cdot \left(t1 \cdot \frac{v}{u}\right)\\ \end{array} \]

Alternative 3
Error	24.64%
Cost	841

\[\begin{array}{l} \mathbf{if}\;t1 \leq -1.4 \cdot 10^{-133} \lor \neg \left(t1 \leq 7.5 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{u} \cdot \left(t1 \cdot \frac{v}{u}\right)\\ \end{array} \]

Alternative 4
Error	24.38%
Cost	840

\[\begin{array}{l} \mathbf{if}\;t1 \leq -1.6 \cdot 10^{-150}:\\ \;\;\;\;\frac{\frac{v}{t1}}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 7.5 \cdot 10^{+67}:\\ \;\;\;\;\frac{-1}{u} \cdot \left(t1 \cdot \frac{v}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 5
Error	24.51%
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -1.4 \cdot 10^{-133} \lor \neg \left(t1 \leq 1.4 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{-u}\\ \end{array} \]

Alternative 6
Error	24.57%
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -1.4 \cdot 10^{-133} \lor \neg \left(t1 \leq 7.5 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{-t1}{u}}{u}\\ \end{array} \]

Alternative 7
Error	32.48%
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -1.12 \cdot 10^{+92} \lor \neg \left(u \leq 2.2 \cdot 10^{+149}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 8
Error	43.22%
Cost	521

\[\begin{array}{l} \mathbf{if}\;u \leq -1.35 \cdot 10^{+156} \lor \neg \left(u \leq 6.4 \cdot 10^{+190}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 9
Error	39.4%
Cost	384

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

Alternative 10
Error	47.86%
Cost	256

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

Reproduce

herbie shell --seed 2023089 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))