Rosa's DopplerBench

Average Error: 28.16% → 2.1%

Time: 11.1s

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 28.16

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

2. Simplified 2.1

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

   [Start] 28.16	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   times-frac [=>] 2.1	\[ \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

3. Final simplification 2.1

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

Alternatives

Alternative 1
Error	23%
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -3.3 \cdot 10^{-112} \lor \neg \left(t1 \leq 1.45 \cdot 10^{-113}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{-u}\\ \end{array} \]

Alternative 2
Error	22.76%
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -3.3 \cdot 10^{-112} \lor \neg \left(t1 \leq 1.85 \cdot 10^{-110}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \end{array} \]

Alternative 3
Error	22.8%
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -3.3 \cdot 10^{-112} \lor \neg \left(t1 \leq 1.1 \cdot 10^{-109}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \end{array} \]

Alternative 4
Error	22.58%
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -2.05 \cdot 10^{-112} \lor \neg \left(t1 \leq 1.85 \cdot 10^{-110}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{-u}\\ \end{array} \]

Alternative 5
Error	32.57%
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -2.1 \cdot 10^{+113} \lor \neg \left(u \leq 4.2 \cdot 10^{+138}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \end{array} \]

Alternative 6
Error	32.33%
Cost	712

\[\begin{array}{l} \mathbf{if}\;u \leq -6 \cdot 10^{+107}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{elif}\;u \leq 5.2 \cdot 10^{+143}:\\ \;\;\;\;\frac{v}{\left(-t1\right) - u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \end{array} \]

Alternative 7
Error	31.92%
Cost	712

\[\begin{array}{l} \mathbf{if}\;u \leq -4.2 \cdot 10^{+109}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{elif}\;u \leq 4.5 \cdot 10^{+144}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \end{array} \]

Alternative 8
Error	2.27%
Cost	704

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

Alternative 9
Error	43.45%
Cost	585

\[\begin{array}{l} \mathbf{if}\;u \leq -2.3 \cdot 10^{+112} \lor \neg \left(u \leq 8.6 \cdot 10^{+149}\right):\\ \;\;\;\;-0.5 \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 10
Error	43.58%
Cost	521

\[\begin{array}{l} \mathbf{if}\;u \leq -6.2 \cdot 10^{+108} \lor \neg \left(u \leq 2 \cdot 10^{+147}\right):\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 11
Error	39.62%
Cost	384

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

Alternative 12
Error	48.24%
Cost	256

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

Reproduce

herbie shell --seed 2023102 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))