Rosa's DopplerBench

Average Error: 28.55% → 2.01%

Time: 9.9s

Precision: binary64

Cost: 768

Math

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

↓

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

FPCore

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

↓

(FPCore (u v t1) :precision binary64 (* (/ (- 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) {
	return (-t1 / (t1 + u)) * (v / (t1 + u));
}

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 = (-t1 / (t1 + u)) * (v / (t1 + u))
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 (-t1 / (t1 + u)) * (v / (t1 + u));
}

Python

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

↓

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

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(Float64(-t1) / Float64(t1 + u)) * Float64(v / Float64(t1 + u)))
end

MATLAB

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

↓

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

Derivation

1. Initial program 28.55

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

2. Simplified 2.01

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

   [Start] 28.55	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   times-frac [=>] 2.01	\[ \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

3. Final simplification 2.01

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

Alternatives

Alternative 1
Error	25.16%
Cost	776

\[\begin{array}{l} \mathbf{if}\;t1 \leq -8.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 30:\\ \;\;\;\;\frac{-t1}{\frac{u \cdot u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 - u}\\ \end{array} \]

Alternative 2
Error	22.15%
Cost	776

\[\begin{array}{l} \mathbf{if}\;t1 \leq -1.75 \cdot 10^{-53}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 95:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 - u}\\ \end{array} \]

Alternative 3
Error	22.26%
Cost	776

\[\begin{array}{l} \mathbf{if}\;t1 \leq -1.4 \cdot 10^{-50}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 290:\\ \;\;\;\;\frac{v \cdot \frac{-t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 - u}\\ \end{array} \]

Alternative 4
Error	22.13%
Cost	776

\[\begin{array}{l} \mathbf{if}\;t1 \leq -2.4 \cdot 10^{-51}:\\ \;\;\;\;\frac{\frac{v}{t1}}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 31:\\ \;\;\;\;\frac{v \cdot \frac{-t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 - u}\\ \end{array} \]

Alternative 5
Error	32.75%
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -4 \cdot 10^{+118} \lor \neg \left(u \leq 2.05 \cdot 10^{+149}\right):\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 6
Error	33.05%
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -2.7 \cdot 10^{+31} \lor \neg \left(u \leq 3.9 \cdot 10^{+153}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 7
Error	2.1%
Cost	704

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

Alternative 8
Error	43.59%
Cost	584

\[\begin{array}{l} \mathbf{if}\;u \leq -1.9 \cdot 10^{+98}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 8 \cdot 10^{+154}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \end{array} \]

Alternative 9
Error	43.7%
Cost	520

\[\begin{array}{l} \mathbf{if}\;u \leq -3.1 \cdot 10^{+98}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 1.55 \cdot 10^{+152}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 10
Error	43.7%
Cost	520

\[\begin{array}{l} \mathbf{if}\;u \leq -2.6 \cdot 10^{+98}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 2.16 \cdot 10^{+148}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u}\\ \end{array} \]

Alternative 11
Error	77.69%
Cost	456

\[\begin{array}{l} \mathbf{if}\;t1 \leq -3.9 \cdot 10^{+136}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 8 \cdot 10^{+120}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \]

Alternative 12
Error	39.35%
Cost	384

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

Alternative 13
Error	39.41%
Cost	384

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

Alternative 14
Error	85.13%
Cost	192

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

Reproduce

herbie shell --seed 2023093 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))