Rosa's DopplerBench

Average Error: 18.2 → 1.5

Time: 14.6s

Precision: binary64

Cost: 768

Math

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

↓

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

FPCore

(FPCore (u v t1) :precision binary64 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))

↓

(FPCore (u v t1) :precision binary64 (/ (/ (- v) (+ t1 u)) (/ (+ t1 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)) / ((t1 + 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)) / ((t1 + 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)) / ((t1 + u) / t1);
}

Python

def code(u, v, t1):
	return (-t1 * v) / ((t1 + u) * (t1 + u))

↓

def code(u, v, t1):
	return (-v / (t1 + u)) / ((t1 + 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(Float64(-v) / Float64(t1 + u)) / Float64(Float64(t1 + 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)) / ((t1 + 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[(N[(t1 + u), $MachinePrecision] / t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.2

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

2. Simplified 17.8

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

   [Start] 18.2	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   rational.json-simplify-2 [=>] 18.2	\[ \frac{\color{blue}{v \cdot \left(-t1\right)}}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   rational.json-simplify-49 [=>] 17.8	\[ \color{blue}{\left(-t1\right) \cdot \frac{v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}} \]

3. Applied egg-rr 1.5

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

4. Final simplification 1.5

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

Alternatives

Alternative 1
Error	15.7
Cost	1168

\[\begin{array}{l} t_1 := \frac{v}{u \cdot -2 + \left(-t1\right)}\\ \mathbf{if}\;u \leq -7.5 \cdot 10^{+63}:\\ \;\;\;\;\frac{-t1}{t1 + u} \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq -2.05 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -2.6 \cdot 10^{-88}:\\ \;\;\;\;\left(-\frac{t1}{u}\right) \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq 1.16 \cdot 10^{-75}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-v}{t1 + u}}{u} \cdot t1\\ \end{array} \]

Alternative 2
Error	15.5
Cost	1168

\[\begin{array}{l} t_1 := \frac{v}{u \cdot -2 + \left(-t1\right)}\\ \mathbf{if}\;u \leq -7.8 \cdot 10^{+66}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{-\left(t1 + u\right)}\\ \mathbf{elif}\;u \leq -1.1 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -1.85 \cdot 10^{-91}:\\ \;\;\;\;\left(-\frac{t1}{u}\right) \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq 2.85 \cdot 10^{-75}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-v}{t1 + u}}{u} \cdot t1\\ \end{array} \]

Alternative 3
Error	15.4
Cost	1168

\[\begin{array}{l} t_1 := \frac{v}{u \cdot -2 + \left(-t1\right)}\\ \mathbf{if}\;u \leq -3.65 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{t1}{\frac{u}{v}}}{-\left(t1 + u\right)}\\ \mathbf{elif}\;u \leq -2.25 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -2.6 \cdot 10^{-88}:\\ \;\;\;\;\left(-\frac{t1}{u}\right) \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq 2.4 \cdot 10^{-75}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-v}{t1 + u}}{\frac{u}{t1}}\\ \end{array} \]

Alternative 4
Error	16.5
Cost	1040

\[\begin{array}{l} t_1 := \frac{-v}{u + t1}\\ t_2 := \left(-\frac{t1}{u}\right) \cdot \frac{v}{u}\\ \mathbf{if}\;u \leq -1.6 \cdot 10^{+90}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq -5.2 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -2.6 \cdot 10^{-88}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq 3.8 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	16.3
Cost	1040

\[\begin{array}{l} t_1 := \frac{v}{u \cdot -2 + \left(-t1\right)}\\ t_2 := \left(-\frac{t1}{u}\right) \cdot \frac{v}{u}\\ \mathbf{if}\;u \leq -1.65 \cdot 10^{+88}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq -3.4 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -2.6 \cdot 10^{-88}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq 1.55 \cdot 10^{+78}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	16.3
Cost	1040

\[\begin{array}{l} t_1 := \frac{v}{u \cdot -2 + \left(-t1\right)}\\ t_2 := \left(-\frac{t1}{u}\right) \cdot \frac{v}{u}\\ \mathbf{if}\;u \leq -4.2 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{-v}{u}}{\frac{u}{t1}}\\ \mathbf{elif}\;u \leq -3.2 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -2.6 \cdot 10^{-88}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq 5 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	13.6
Cost	776

\[\begin{array}{l} \mathbf{if}\;t1 \leq -5.1 \cdot 10^{-90}:\\ \;\;\;\;\frac{-t1}{t1 + u} \cdot \frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 7.2 \cdot 10^{-44}:\\ \;\;\;\;\left(-\frac{t1}{u}\right) \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 + \left(-t1\right)}\\ \end{array} \]

Alternative 8
Error	1.3
Cost	768

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

Alternative 9
Error	3.1
Cost	704

\[\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)} \]

Alternative 10
Error	27.3
Cost	520

\[\begin{array}{l} t_1 := -\frac{v}{u}\\ \mathbf{if}\;u \leq -3.2 \cdot 10^{+191}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 3 \cdot 10^{+212}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	24.7
Cost	384

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

Alternative 12
Error	30.3
Cost	256

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

Alternative 13
Error	54.6
Cost	192

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

Reproduce

herbie shell --seed 2023069 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))