Rosa's DopplerBench

Average Error: 17.7 → 1.3

Time: 13.6s

Precision: binary64

Cost: 1097

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t1 \leq -9.2 \cdot 10^{-133} \lor \neg \left(t1 \leq 4 \cdot 10^{-150}\right):\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \frac{-1}{\frac{t1}{t1 + u}}}\\ \end{array} \]

FPCore

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

↓

(FPCore (u v t1)
 :precision binary64
 (if (or (<= t1 -9.2e-133) (not (<= t1 4e-150)))
   (/ (/ v (+ t1 u)) (- -1.0 (/ u t1)))
   (/ v (* (+ t1 u) (/ -1.0 (/ t1 (+ 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 tmp;
	if ((t1 <= -9.2e-133) || !(t1 <= 4e-150)) {
		tmp = (v / (t1 + u)) / (-1.0 - (u / t1));
	} else {
		tmp = v / ((t1 + u) * (-1.0 / (t1 / (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) :: tmp
    if ((t1 <= (-9.2d-133)) .or. (.not. (t1 <= 4d-150))) then
        tmp = (v / (t1 + u)) / ((-1.0d0) - (u / t1))
    else
        tmp = v / ((t1 + u) * ((-1.0d0) / (t1 / (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 tmp;
	if ((t1 <= -9.2e-133) || !(t1 <= 4e-150)) {
		tmp = (v / (t1 + u)) / (-1.0 - (u / t1));
	} else {
		tmp = v / ((t1 + u) * (-1.0 / (t1 / (t1 + 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 <= -9.2e-133) or not (t1 <= 4e-150):
		tmp = (v / (t1 + u)) / (-1.0 - (u / t1))
	else:
		tmp = v / ((t1 + u) * (-1.0 / (t1 / (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)
	tmp = 0.0
	if ((t1 <= -9.2e-133) || !(t1 <= 4e-150))
		tmp = Float64(Float64(v / Float64(t1 + u)) / Float64(-1.0 - Float64(u / t1)));
	else
		tmp = Float64(v / Float64(Float64(t1 + u) * Float64(-1.0 / Float64(t1 / 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)
	tmp = 0.0;
	if ((t1 <= -9.2e-133) || ~((t1 <= 4e-150)))
		tmp = (v / (t1 + u)) / (-1.0 - (u / t1));
	else
		tmp = v / ((t1 + u) * (-1.0 / (t1 / (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_] := If[Or[LessEqual[t1, -9.2e-133], N[Not[LessEqual[t1, 4e-150]], $MachinePrecision]], N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v / N[(N[(t1 + u), $MachinePrecision] * N[(-1.0 / N[(t1 / N[(t1 + u), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t1 < -9.2000000000000001e-133 or 4.00000000000000003e-150 < t1

   1. Initial program 18.8

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

   2. Simplified 0.2

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

      [Start] 18.8	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      *-commutative [=>] 18.8	\[ \frac{\color{blue}{v \cdot \left(-t1\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      associate-/l* [=>] 15.8	\[ \color{blue}{\frac{v}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}} \]
      associate-*r/ [<=] 3.0	\[ \frac{v}{\color{blue}{\left(t1 + u\right) \cdot \frac{t1 + u}{-t1}}} \]
      associate-/r* [=>] 0.2	\[ \color{blue}{\frac{\frac{v}{t1 + u}}{\frac{t1 + u}{-t1}}} \]
      neg-mul-1 [=>] 0.2	\[ \frac{\frac{v}{t1 + u}}{\frac{t1 + u}{\color{blue}{-1 \cdot t1}}} \]
      associate-/l/ [<=] 0.2	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{\frac{t1 + u}{t1}}{-1}}} \]
      metadata-eval [<=] 0.2	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{0 - 1}}} \]
      mul0-lft [<=] 2.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{0 \cdot \frac{t1 + u}{t1}} - 1}} \]
      associate-*r/ [=>] 0.2	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{0 \cdot \left(t1 + u\right)}{t1}} - 1}} \]
      mul0-lft [=>] 0.2	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{0}}{t1} - 1}} \]
      *-inverses [<=] 0.2	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{0}{t1} - \color{blue}{\frac{t1}{t1}}}} \]
      div-sub [<=] 0.2	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{0 - t1}{t1}}}} \]
      neg-sub0 [<=] 0.2	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{-t1}}{t1}}} \]
      neg-mul-1 [=>] 0.2	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{-1 \cdot t1}}{t1}}} \]
      *-commutative [=>] 0.2	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{t1 \cdot -1}}{t1}}} \]
      associate-/l* [=>] 0.2	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{t1}{\frac{t1}{-1}}}}} \]
      associate-/l* [<=] 0.2	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{\frac{t1 + u}{t1} \cdot \frac{t1}{-1}}{t1}}} \]
      *-commutative [=>] 0.2	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1}{-1} \cdot \frac{t1 + u}{t1}}}{t1}} \]
      times-frac [<=] 14.0	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1 \cdot \left(t1 + u\right)}{-1 \cdot t1}}}{t1}} \]
      neg-mul-1 [<=] 14.0	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 \cdot \left(t1 + u\right)}{\color{blue}{-t1}}}{t1}} \]
      associate-/l* [=>] 0.2	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1}{\frac{-t1}{t1 + u}}}}{t1}} \]

   if -9.2000000000000001e-133 < t1 < 4.00000000000000003e-150

   1. Initial program 14.6

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

   2. Simplified 14.7

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

      [Start] 14.6	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      *-commutative [=>] 14.6	\[ \frac{\color{blue}{v \cdot \left(-t1\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
      associate-/l* [=>] 14.7	\[ \color{blue}{\frac{v}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}} \]

   3. Applied egg-rr 4.3

      \[\leadsto \frac{v}{\color{blue}{\frac{-1}{\frac{t1}{t1 + u}} \cdot \left(t1 + u\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t1 \leq -9.2 \cdot 10^{-133} \lor \neg \left(t1 \leq 4 \cdot 10^{-150}\right):\\ \;\;\;\;\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \frac{-1}{\frac{t1}{t1 + u}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	18.1
Cost	1040

\[\begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ t_2 := t1 \cdot \frac{-v}{u \cdot u}\\ \mathbf{if}\;u \leq -1.85 \cdot 10^{-39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq -2.75 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -2.4 \cdot 10^{-120}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{elif}\;u \leq 4.3 \cdot 10^{-44}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	16.1
Cost	1040

\[\begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ t_2 := \frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{if}\;u \leq -4.2 \cdot 10^{-39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq -7.3 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -2.4 \cdot 10^{-120}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{elif}\;u \leq 1.7 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	16.2
Cost	1040

\[\begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;u \leq -1.04 \cdot 10^{-38}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \mathbf{elif}\;u \leq -6 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -2.4 \cdot 10^{-120}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{elif}\;u \leq 8.2 \cdot 10^{-44}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \end{array} \]

Alternative 4
Error	15.8
Cost	1040

\[\begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;u \leq -6.4 \cdot 10^{-40}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq -1.4 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -2.4 \cdot 10^{-120}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{elif}\;u \leq 1.25 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \end{array} \]

Alternative 5
Error	15.8
Cost	1040

\[\begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;u \leq -6.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{t1 - u}\\ \mathbf{elif}\;u \leq -2.55 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -2.4 \cdot 10^{-120}:\\ \;\;\;\;v \cdot \frac{\frac{-t1}{t1 + u}}{u}\\ \mathbf{elif}\;u \leq 1.2 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \end{array} \]

Alternative 6
Error	14.0
Cost	905

\[\begin{array}{l} \mathbf{if}\;u \leq -5.8 \cdot 10^{-95} \lor \neg \left(u \leq 5.8 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]

Alternative 7
Error	14.9
Cost	904

\[\begin{array}{l} t_1 := \frac{v}{t1 + u}\\ \mathbf{if}\;u \leq -1.12 \cdot 10^{-125}:\\ \;\;\;\;\frac{-t1}{\frac{u}{t_1}}\\ \mathbf{elif}\;u \leq 4.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{-t1}{u}\\ \end{array} \]

Alternative 8
Error	16.1
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -9.2 \cdot 10^{+33} \lor \neg \left(t1 \leq 4.6 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \end{array} \]

Alternative 9
Error	21.4
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -8.6 \cdot 10^{+25} \lor \neg \left(u \leq 1.55 \cdot 10^{+112}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 10
Error	21.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;u \leq -9.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{elif}\;u \leq 9.5 \cdot 10^{+112}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \end{array} \]

Alternative 11
Error	20.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;u \leq -3.1 \cdot 10^{+64}:\\ \;\;\;\;\frac{t1}{u \cdot \frac{u}{v}}\\ \mathbf{elif}\;u \leq 1.35 \cdot 10^{+118}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \end{array} \]

Alternative 12
Error	1.5
Cost	704

\[\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}} \]

Alternative 13
Error	29.1
Cost	520

\[\begin{array}{l} \mathbf{if}\;u \leq -3.1 \cdot 10^{+43}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 1.75 \cdot 10^{+78}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 14
Error	29.1
Cost	520

\[\begin{array}{l} \mathbf{if}\;u \leq -8.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{elif}\;u \leq 1.6 \cdot 10^{+78}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 15
Error	29.1
Cost	520

\[\begin{array}{l} \mathbf{if}\;u \leq -8.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{elif}\;u \leq 1.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 16
Error	25.2
Cost	384

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

Alternative 17
Error	54.0
Cost	192

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

Reproduce

herbie shell --seed 2023018 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))