Rosa's DopplerBench

Average Error: 18.4 → 1.3

Time: 11.6s

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 18.4

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

2. Simplified 1.3

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

   [Start] 18.4	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   times-frac [=>] 1.3	\[ \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

3. Final simplification 1.3

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

Alternatives

Alternative 1
Error	15.6
Cost	1042

\[\begin{array}{l} \mathbf{if}\;t1 \leq -1 \cdot 10^{-40} \lor \neg \left(t1 \leq -3.2 \cdot 10^{-94}\right) \land \left(t1 \leq -5.7 \cdot 10^{-148} \lor \neg \left(t1 \leq 9 \cdot 10^{-61}\right)\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \end{array} \]

Alternative 2
Error	14.1
Cost	1042

\[\begin{array}{l} \mathbf{if}\;t1 \leq -1.5 \cdot 10^{-31} \lor \neg \left(t1 \leq -3.75 \cdot 10^{-94}\right) \land \left(t1 \leq -1.52 \cdot 10^{-146} \lor \neg \left(t1 \leq 8.2 \cdot 10^{-61}\right)\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{t1}{-u}\\ \end{array} \]

Alternative 3
Error	15.7
Cost	1040

\[\begin{array}{l} \mathbf{if}\;u \leq -11200000000000:\\ \;\;\;\;\frac{-1}{u} \cdot \left(t1 \cdot \frac{v}{u}\right)\\ \mathbf{elif}\;u \leq 6.6 \cdot 10^{-104}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{elif}\;u \leq 1.45 \cdot 10^{-44}:\\ \;\;\;\;\frac{v}{\frac{u \cdot u}{-t1}}\\ \mathbf{elif}\;u \leq 520000000:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \end{array} \]

Alternative 4
Error	3.4
Cost	836

\[\begin{array}{l} \mathbf{if}\;u \leq -8.9 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{t1}{u}}{\frac{-u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \left(-2 - \frac{u}{t1}\right) - t1}\\ \end{array} \]

Alternative 5
Error	21.0
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -1.8 \cdot 10^{+117} \lor \neg \left(u \leq 2.4 \cdot 10^{+99}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 6
Error	21.4
Cost	713

\[\begin{array}{l} \mathbf{if}\;t1 \leq -2.2 \cdot 10^{-217} \lor \neg \left(t1 \leq 8.5 \cdot 10^{-156}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot \frac{u}{t1}}\\ \end{array} \]

Alternative 7
Error	1.4
Cost	704

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

Alternative 8
Error	28.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;u \leq -6.2 \cdot 10^{+117}:\\ \;\;\;\;\frac{v}{u} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 9
Error	28.8
Cost	388

\[\begin{array}{l} \mathbf{if}\;u \leq -6.2 \cdot 10^{+117}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 10
Error	30.0
Cost	256

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

Reproduce

herbie shell --seed 2023045 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))