Rosa's DopplerBench

Average Error: 28.66% → 2.4%

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 28.66

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

2. Simplified 2.4

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

   [Start] 28.66	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   times-frac [=>] 2.4	\[ \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

3. Final simplification 2.4

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

Alternatives

Alternative 1
Error	25.07%
Cost	1168

\[\begin{array}{l} t_1 := \frac{-v}{t1}\\ t_2 := \frac{\frac{t1}{u}}{\frac{\left(-t1\right) - u}{v}}\\ \mathbf{if}\;t1 \leq -7.5 \cdot 10^{+178}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -1.35 \cdot 10^{+116}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq -5.2 \cdot 10^{-68}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 2.15 \cdot 10^{-30}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	25.38%
Cost	1040

\[\begin{array}{l} t_1 := \frac{-v}{t1}\\ \mathbf{if}\;t1 \leq -7.5 \cdot 10^{+178}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -1 \cdot 10^{+116}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{elif}\;t1 \leq -1.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 1.02 \cdot 10^{-41}:\\ \;\;\;\;\frac{t1 \cdot \left(-\frac{v}{u}\right)}{u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	23.3%
Cost	776

\[\begin{array}{l} \mathbf{if}\;t1 \leq -4 \cdot 10^{-67}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 2.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{-t1}{u \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 4
Error	22.41%
Cost	776

\[\begin{array}{l} \mathbf{if}\;t1 \leq -1.12 \cdot 10^{-69}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 7.1 \cdot 10^{-34}:\\ \;\;\;\;\frac{t1}{u} \cdot \left(-\frac{v}{u}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 5
Error	22.42%
Cost	776

\[\begin{array}{l} \mathbf{if}\;t1 \leq -2.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 8 \cdot 10^{-30}:\\ \;\;\;\;\frac{t1 \cdot \left(-\frac{v}{u}\right)}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 6
Error	33.16%
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -5.2 \cdot 10^{+162} \lor \neg \left(u \leq 2.4 \cdot 10^{+94}\right):\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 7
Error	33.67%
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -4.6 \cdot 10^{+28} \lor \neg \left(u \leq 2.8 \cdot 10^{+92}\right):\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 8
Error	33.24%
Cost	712

\[\begin{array}{l} \mathbf{if}\;u \leq -6.8 \cdot 10^{+162}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{elif}\;u \leq 1.15 \cdot 10^{+93}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{u}\\ \end{array} \]

Alternative 9
Error	2.56%
Cost	704

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

Alternative 10
Error	43.21%
Cost	521

\[\begin{array}{l} \mathbf{if}\;u \leq -6.8 \cdot 10^{+140} \lor \neg \left(u \leq 7.5 \cdot 10^{+148}\right):\\ \;\;\;\;-\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 11
Error	39.48%
Cost	384

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

Alternative 12
Error	48.56%
Cost	256

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

Alternative 13
Error	85.93%
Cost	192

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

Reproduce

herbie shell --seed 2023101 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))