Rosa's DopplerBench

Average Error: 18.4 → 1.5

Time: 12.0s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

(FPCore (u v t1) :precision binary64 (/ (/ v (+ t1 u)) (- -1.0 (/ u t1))))

C

double code(double u, double v, double t1) {
	return (-t1 * v) / ((t1 + u) * (t1 + u));
}

↓

double code(double u, double v, double t1) {
	return (v / (t1 + u)) / (-1.0 - (u / t1));
}

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
    code = (v / (t1 + u)) / ((-1.0d0) - (u / t1))
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) {
	return (v / (t1 + u)) / (-1.0 - (u / t1));
}

Python

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

↓

def code(u, v, t1):
	return (v / (t1 + u)) / (-1.0 - (u / t1))

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)
	return Float64(Float64(v / Float64(t1 + u)) / Float64(-1.0 - Float64(u / t1)))
end

MATLAB

function tmp = code(u, v, t1)
	tmp = (-t1 * v) / ((t1 + u) * (t1 + u));
end

↓

function tmp = code(u, v, t1)
	tmp = (v / (t1 + u)) / (-1.0 - (u / t1));
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_] := N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[(u / t1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.4

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

2. Simplified 1.5

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

   [Start] 18.4	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   *-commutative [=>] 18.4	\[ \frac{\color{blue}{v \cdot \left(-t1\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   associate-/l* [=>] 16.1	\[ \color{blue}{\frac{v}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}} \]
   associate-*r/ [<=] 3.4	\[ \frac{v}{\color{blue}{\left(t1 + u\right) \cdot \frac{t1 + u}{-t1}}} \]
   associate-/r* [=>] 1.5	\[ \color{blue}{\frac{\frac{v}{t1 + u}}{\frac{t1 + u}{-t1}}} \]
   neg-mul-1 [=>] 1.5	\[ \frac{\frac{v}{t1 + u}}{\frac{t1 + u}{\color{blue}{-1 \cdot t1}}} \]
   associate-/l/ [<=] 1.5	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{\frac{t1 + u}{t1}}{-1}}} \]
   metadata-eval [<=] 1.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{0 - 1}}} \]
   mul0-lft [<=] 9.1	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{0 \cdot \frac{t1 + u}{t1}} - 1}} \]
   associate-*r/ [=>] 1.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{0 \cdot \left(t1 + u\right)}{t1}} - 1}} \]
   mul0-lft [=>] 1.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{0}}{t1} - 1}} \]
   *-inverses [<=] 1.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{0}{t1} - \color{blue}{\frac{t1}{t1}}}} \]
   div-sub [<=] 1.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{0 - t1}{t1}}}} \]
   neg-sub0 [<=] 1.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{-t1}}{t1}}} \]
   neg-mul-1 [=>] 1.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{-1 \cdot t1}}{t1}}} \]
   *-commutative [=>] 1.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\frac{\color{blue}{t1 \cdot -1}}{t1}}} \]
   associate-/l* [=>] 1.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 + u}{t1}}{\color{blue}{\frac{t1}{\frac{t1}{-1}}}}} \]
   associate-/l* [<=] 1.5	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{\frac{t1 + u}{t1} \cdot \frac{t1}{-1}}{t1}}} \]
   *-commutative [=>] 1.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1}{-1} \cdot \frac{t1 + u}{t1}}}{t1}} \]
   times-frac [<=] 16.1	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1 \cdot \left(t1 + u\right)}{-1 \cdot t1}}}{t1}} \]
   neg-mul-1 [<=] 16.1	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 \cdot \left(t1 + u\right)}{\color{blue}{-t1}}}{t1}} \]
   associate-/l* [=>] 1.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1}{\frac{-t1}{t1 + u}}}}{t1}} \]
   associate-/r/ [=>] 1.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1}{-t1} \cdot \left(t1 + u\right)}}{t1}} \]
   neg-mul-1 [=>] 1.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1}{\color{blue}{-1 \cdot t1}} \cdot \left(t1 + u\right)}{t1}} \]
   *-commutative [=>] 1.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1}{\color{blue}{t1 \cdot -1}} \cdot \left(t1 + u\right)}{t1}} \]
   associate-/r* [=>] 1.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{\frac{t1}{t1}}{-1}} \cdot \left(t1 + u\right)}{t1}} \]
   *-inverses [=>] 1.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{\color{blue}{1}}{-1} \cdot \left(t1 + u\right)}{t1}} \]
   metadata-eval [=>] 1.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{-1} \cdot \left(t1 + u\right)}{t1}} \]
   neg-mul-1 [<=] 1.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{-\left(t1 + u\right)}}{t1}} \]
   distribute-neg-in [=>] 1.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(-t1\right) + \left(-u\right)}}{t1}} \]
   sub-neg [<=] 1.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\left(-t1\right) - u}}{t1}} \]
   div-sub [=>] 1.5	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\frac{-t1}{t1} - \frac{u}{t1}}} \]
   neg-sub0 [=>] 1.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{0 - t1}}{t1} - \frac{u}{t1}} \]
   div-sub [=>] 1.5	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{\left(\frac{0}{t1} - \frac{t1}{t1}\right)} - \frac{u}{t1}} \]
   mul0-lft [<=] 1.5	\[ \frac{\frac{v}{t1 + u}}{\left(\frac{\color{blue}{0 \cdot \left(t1 + u\right)}}{t1} - \frac{t1}{t1}\right) - \frac{u}{t1}} \]
   associate-*r/ [<=] 9.1	\[ \frac{\frac{v}{t1 + u}}{\left(\color{blue}{0 \cdot \frac{t1 + u}{t1}} - \frac{t1}{t1}\right) - \frac{u}{t1}} \]
   mul0-lft [=>] 1.5	\[ \frac{\frac{v}{t1 + u}}{\left(\color{blue}{0} - \frac{t1}{t1}\right) - \frac{u}{t1}} \]
   *-inverses [=>] 1.5	\[ \frac{\frac{v}{t1 + u}}{\left(0 - \color{blue}{1}\right) - \frac{u}{t1}} \]
   metadata-eval [=>] 1.5	\[ \frac{\frac{v}{t1 + u}}{\color{blue}{-1} - \frac{u}{t1}} \]

3. Final simplification 1.5

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

Alternatives

Alternative 1
Error	3.1
Cost	836

\[\begin{array}{l} \mathbf{if}\;u \leq -4.1 \cdot 10^{+245}:\\ \;\;\;\;\frac{-1}{u} \cdot \left(t1 \cdot \frac{v}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\ \end{array} \]

Alternative 2
Error	16.1
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -0.155 \lor \neg \left(t1 \leq 9.5 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{\frac{u \cdot u}{v}}\\ \end{array} \]

Alternative 3
Error	14.9
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -2 \lor \neg \left(t1 \leq 1.3 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{u}}{u}\\ \end{array} \]

Alternative 4
Error	14.0
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -0.0275 \lor \neg \left(t1 \leq 1.32 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u} \cdot \frac{t1}{u}\\ \end{array} \]

Alternative 5
Error	13.9
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -5.4 \lor \neg \left(t1 \leq 4.6 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{-u} \cdot \frac{t1}{u}\\ \end{array} \]

Alternative 6
Error	13.8
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -3.25 \lor \neg \left(t1 \leq 4.8 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{-v}{t1 + u \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{-u}}{\frac{u}{v}}\\ \end{array} \]

Alternative 7
Error	20.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -4.7 \cdot 10^{+76} \lor \neg \left(u \leq 1.8 \cdot 10^{+115}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 8
Error	27.2
Cost	585

\[\begin{array}{l} \mathbf{if}\;u \leq -1.3 \cdot 10^{+158} \lor \neg \left(u \leq 6 \cdot 10^{+155}\right):\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]

Alternative 9
Error	27.1
Cost	585

\[\begin{array}{l} \mathbf{if}\;u \leq -3.2 \cdot 10^{+159} \lor \neg \left(u \leq 2.5 \cdot 10^{+155}\right):\\ \;\;\;\;\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]

Alternative 10
Error	27.2
Cost	521

\[\begin{array}{l} \mathbf{if}\;u \leq -2 \cdot 10^{+159} \lor \neg \left(u \leq 4.7 \cdot 10^{+155}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]

Alternative 11
Error	24.9
Cost	384

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

Alternative 12
Error	30.2
Cost	256

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

Reproduce

herbie shell --seed 2022356 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))