Rosa's DopplerBench

Average Error: 17.9 → 1.5

Time: 12.3s

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 17.9

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

2. Simplified 1.5

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

   [Start] 17.9	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   *-commutative [=>] 17.9	\[ \frac{\color{blue}{v \cdot \left(-t1\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   associate-/l* [=>] 15.8	\[ \color{blue}{\frac{v}{\frac{\left(t1 + u\right) \cdot \left(t1 + u\right)}{-t1}}} \]
   associate-*r/ [<=] 3.5	\[ \frac{v}{\color{blue}{\left(t1 + u\right) \cdot \frac{t1 + u}{-t1}}} \]
   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 [<=] 16.0	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1 \cdot \left(t1 + u\right)}{-1 \cdot t1}}}{t1}} \]
   neg-mul-1 [<=] 16.0	\[ \frac{\frac{v}{t1 + u}}{\frac{\frac{t1 \cdot \left(t1 + u\right)}{\color{blue}{-t1}}}{t1}} \]
   associate-/l* [=>] 1.5	\[ \frac{\frac{v}{t1 + u}}{\frac{\color{blue}{\frac{t1}{\frac{-t1}{t1 + u}}}}{t1}} \]

3. Final simplification 1.5

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

Alternatives

Alternative 1
Error	13.9
Cost	1172

\[\begin{array}{l} t_1 := \frac{t1}{u} \cdot \frac{-v}{u}\\ t_2 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -7.4 \cdot 10^{-9}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq -1.05 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -5.4 \cdot 10^{-53}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 2.8 \cdot 10^{-239}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{elif}\;t1 \leq 3.1 \cdot 10^{-72}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	15.1
Cost	1168

\[\begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ t_2 := \frac{t1}{u} \cdot \frac{-v}{t1 + u}\\ \mathbf{if}\;u \leq -1.4 \cdot 10^{+77}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq -3.5 \cdot 10^{-98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -8.5 \cdot 10^{-128}:\\ \;\;\;\;\frac{v}{\left(t1 - u\right) \cdot \frac{t1 + u}{t1}}\\ \mathbf{elif}\;u \leq 9.5 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	13.7
Cost	1040

\[\begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -7.2 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -1.9 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{v}{u}}{\frac{-u}{t1}}\\ \mathbf{elif}\;t1 \leq -8.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 7.2 \cdot 10^{-67}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	13.7
Cost	1040

\[\begin{array}{l} t_1 := \frac{v}{u \cdot -2 - t1}\\ \mathbf{if}\;t1 \leq -8 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -1.15 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{v}{u}}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq -5.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;t1 \leq 1.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	15.2
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -9 \cdot 10^{-59} \lor \neg \left(t1 \leq 3.1 \cdot 10^{-71}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \end{array} \]

Alternative 6
Error	13.7
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -7.8 \cdot 10^{-9} \lor \neg \left(t1 \leq 6.1 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \]

Alternative 7
Error	22.7
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -9 \cdot 10^{+92} \lor \neg \left(u \leq 1.28 \cdot 10^{+183}\right):\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 8
Error	21.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -1.02 \cdot 10^{+93} \lor \neg \left(u \leq 1.3 \cdot 10^{+113}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 9
Error	20.5
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -2.9 \cdot 10^{+95} \lor \neg \left(u \leq 2.4 \cdot 10^{+114}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]

Alternative 10
Error	27.6
Cost	585

\[\begin{array}{l} \mathbf{if}\;u \leq -1.2 \cdot 10^{+149} \lor \neg \left(u \leq 3.9 \cdot 10^{+185}\right):\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 11
Error	27.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;u \leq -3 \cdot 10^{+149}:\\ \;\;\;\;v \cdot \frac{-0.5}{u}\\ \mathbf{elif}\;u \leq 1.95 \cdot 10^{+186}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \end{array} \]

Alternative 12
Error	27.6
Cost	521

\[\begin{array}{l} \mathbf{if}\;u \leq -1 \cdot 10^{+150} \lor \neg \left(u \leq 1.7 \cdot 10^{+188}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 13
Error	30.8
Cost	256

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

Alternative 14
Error	54.4
Cost	192

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

Reproduce

herbie shell --seed 2023054 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))