Rosa's DopplerBench

Average Error: 18.6 → 1.3

Time: 6.8s

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 18.6

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

2. Simplified 1.3

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

3. Final simplification 1.3

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

Alternatives

Alternative 1
Error	14.5
Cost	776

\[\begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -3.3728445599631048 \cdot 10^{-99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 2.57114464691845 \cdot 10^{-77}:\\ \;\;\;\;-\frac{\frac{t1}{\frac{u}{v}}}{u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	14.5
Cost	776

\[\begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -3.3728445599631048 \cdot 10^{-99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 2.57114464691845 \cdot 10^{-77}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	14.3
Cost	776

\[\begin{array}{l} t_1 := \frac{-v}{t1 + u \cdot 2}\\ \mathbf{if}\;t1 \leq -3.3728445599631048 \cdot 10^{-99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 2.57114464691845 \cdot 10^{-77}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	21.6
Cost	712

\[\begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -3.3728445599631048 \cdot 10^{-99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 1.435385289612417 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{t1}{u}}{\frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	20.9
Cost	712

\[\begin{array}{l} t_1 := t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{if}\;u \leq -5.94048784708551 \cdot 10^{+117}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 1.4568147945148145 \cdot 10^{+47}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	27.2
Cost	584

\[\begin{array}{l} t_1 := \frac{v}{u} \cdot -0.5\\ \mathbf{if}\;u \leq -3.1952352815646294 \cdot 10^{+164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 4.5230609673672734 \cdot 10^{+188}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	27.2
Cost	520

\[\begin{array}{l} t_1 := \frac{-v}{u}\\ \mathbf{if}\;u \leq -3.1952352815646294 \cdot 10^{+164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 4.5230609673672734 \cdot 10^{+188}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	24.6
Cost	384

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

Alternative 9
Error	53.7
Cost	256

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

Reproduce

herbie shell --seed 2022228 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))