Rosa's DopplerBench

Average Error: 18.1 → 1.4

Time: 19.5s

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 18.1

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

2. Simplified 1.4

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

   [Start] 18.1	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   times-frac [=>] 1.4	\[ \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

3. Final simplification 1.4

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

Alternatives

Alternative 1
Error	14.4
Cost	1168

\[\begin{array}{l} t_1 := \frac{\frac{v}{t1 + u}}{\frac{-u}{t1}}\\ \mathbf{if}\;u \leq -3.15 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -3.3 \cdot 10^{-21}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;u \leq -5.7 \cdot 10^{-64}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{elif}\;u \leq 5.6 \cdot 10^{-39}:\\ \;\;\;\;\frac{-v}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	15.1
Cost	1104

\[\begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+40}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{elif}\;u \leq -3.3 \cdot 10^{-21}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;u \leq -1.52 \cdot 10^{-66}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{elif}\;u \leq 5.8 \cdot 10^{-39}:\\ \;\;\;\;\frac{-v}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\left(t1 - u\right) \cdot \frac{u}{v}}\\ \end{array} \]

Alternative 3
Error	15.1
Cost	1104

\[\begin{array}{l} \mathbf{if}\;u \leq -1.35 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{v}{u}}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;u \leq -4.3 \cdot 10^{-21}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;u \leq -5.9 \cdot 10^{-64}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{elif}\;u \leq 4.6 \cdot 10^{-39}:\\ \;\;\;\;\frac{-v}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\left(t1 - u\right) \cdot \frac{u}{v}}\\ \end{array} \]

Alternative 4
Error	15.2
Cost	1104

\[\begin{array}{l} \mathbf{if}\;u \leq -2.3 \cdot 10^{+39}:\\ \;\;\;\;\frac{-t1}{t1 + u} \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq -4.2 \cdot 10^{-21}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;u \leq -1.2 \cdot 10^{-69}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{elif}\;u \leq 5.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{-v}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\left(t1 - u\right) \cdot \frac{u}{v}}\\ \end{array} \]

Alternative 5
Error	17.7
Cost	1041

\[\begin{array}{l} t_1 := \frac{-t1}{\frac{u \cdot u}{v}}\\ \mathbf{if}\;u \leq -1.4 \cdot 10^{+41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -2.6 \cdot 10^{-21}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;u \leq -1.35 \cdot 10^{-64} \lor \neg \left(u \leq 42000\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 - u}\\ \end{array} \]

Alternative 6
Error	15.9
Cost	1041

\[\begin{array}{l} t_1 := \frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{if}\;u \leq -9.8 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -8.2 \cdot 10^{-21}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;u \leq -5.9 \cdot 10^{-64} \lor \neg \left(u \leq 11000\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 - u}\\ \end{array} \]

Alternative 7
Error	15.9
Cost	1040

\[\begin{array}{l} t_1 := \frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{if}\;u \leq -2.2 \cdot 10^{+43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -3.9 \cdot 10^{-19}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;u \leq -5.8 \cdot 10^{-64}:\\ \;\;\;\;v \cdot \frac{-t1}{u \cdot u}\\ \mathbf{elif}\;u \leq 280:\\ \;\;\;\;\frac{-v}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	15.9
Cost	1040

\[\begin{array}{l} t_1 := \frac{t1}{u} \cdot \frac{-v}{u}\\ \mathbf{if}\;u \leq -3.1 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -9.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{elif}\;u \leq -2.65 \cdot 10^{-65}:\\ \;\;\;\;\left(-v\right) \cdot \frac{\frac{t1}{u}}{u}\\ \mathbf{elif}\;u \leq 260000:\\ \;\;\;\;\frac{-v}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	20.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -6.5 \cdot 10^{+160} \lor \neg \left(u \leq 5.8 \cdot 10^{+134}\right):\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 - u}\\ \end{array} \]

Alternative 10
Error	20.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -1.22 \cdot 10^{+159} \lor \neg \left(u \leq 1.04 \cdot 10^{+135}\right):\\ \;\;\;\;\frac{v}{\frac{u \cdot u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 - u}\\ \end{array} \]

Alternative 11
Error	1.5
Cost	704

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

Alternative 12
Error	1.5
Cost	704

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

Alternative 13
Error	49.2
Cost	457

\[\begin{array}{l} \mathbf{if}\;t1 \leq -1.95 \cdot 10^{+101} \lor \neg \left(t1 \leq 1.1 \cdot 10^{+129}\right):\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 14
Error	28.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;u \leq -1.2 \cdot 10^{+171}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 15
Error	28.5
Cost	388

\[\begin{array}{l} \mathbf{if}\;u \leq -9.2 \cdot 10^{+169}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 16
Error	28.6
Cost	388

\[\begin{array}{l} \mathbf{if}\;u \leq -1.75 \cdot 10^{+171}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 17
Error	24.7
Cost	384

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

Alternative 18
Error	24.7
Cost	384

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

Alternative 19
Error	54.4
Cost	192

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

Reproduce

herbie shell --seed 2023038 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))