Rosa's DopplerBench

Average Error: 18.3 → 1.4

Time: 25.9s

Precision: binary64

Cost: 832

Math

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

↓

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

FPCore

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

↓

(FPCore (u v t1)
 :precision binary64
 (* (/ -1.0 (+ t1 u)) (* (/ v (+ 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 (-1.0 / (t1 + u)) * ((v / (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 = ((-1.0d0) / (t1 + u)) * ((v / (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 (-1.0 / (t1 + u)) * ((v / (t1 + u)) * t1);
}

Python

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

↓

def code(u, v, t1):
	return (-1.0 / (t1 + u)) * ((v / (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(-1.0 / Float64(t1 + u)) * Float64(Float64(v / 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 = (-1.0 / (t1 + u)) * ((v / (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[(-1.0 / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * N[(N[(v / N[(t1 + u), $MachinePrecision]), $MachinePrecision] * t1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.3

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

2. Applied egg-rr 11.3

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

3. Applied egg-rr 1.4

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

Alternatives

Alternative 1
Error	8.7
Cost	1164

\[\begin{array}{l} t_1 := \left(-t1\right) \cdot \frac{\frac{v}{t1 + u}}{t1 + u}\\ \mathbf{if}\;t1 \leq -1.55 \cdot 10^{-232}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 1.2 \cdot 10^{-169}:\\ \;\;\;\;\frac{\frac{v}{u} \cdot t1}{-u}\\ \mathbf{elif}\;t1 \leq 3.6 \cdot 10^{+72}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 2
Error	15.4
Cost	1100

\[\begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -1.6 \cdot 10^{+38}:\\ \;\;\;\;\frac{-t1}{t1 + u} \cdot \frac{v}{t1}\\ \mathbf{elif}\;t1 \leq -1.55 \cdot 10^{-24}:\\ \;\;\;\;t_1 \cdot \frac{t1}{u}\\ \mathbf{elif}\;t1 \leq -3.5 \cdot 10^{-145}:\\ \;\;\;\;\frac{-1}{t1} \cdot \frac{t1 \cdot v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 3 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{v}{u} \cdot t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	15.3
Cost	1100

\[\begin{array}{l} \mathbf{if}\;t1 \leq -2.5 \cdot 10^{+36}:\\ \;\;\;\;\frac{-t1}{t1 + u} \cdot \frac{v}{t1}\\ \mathbf{elif}\;t1 \leq -2 \cdot 10^{-27}:\\ \;\;\;\;\frac{-1}{u} \cdot \left(\frac{v}{t1 + u} \cdot t1\right)\\ \mathbf{elif}\;t1 \leq -3.1 \cdot 10^{-145}:\\ \;\;\;\;\frac{-1}{t1} \cdot \frac{t1 \cdot v}{t1 + u}\\ \mathbf{elif}\;t1 \leq 3.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{v}{u} \cdot t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 4
Error	17.1
Cost	1040

\[\begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ t_2 := \frac{t1}{u \cdot u} \cdot \left(-v\right)\\ \mathbf{if}\;t1 \leq -2 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -2.4 \cdot 10^{-21}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq -3.5 \cdot 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 3.85 \cdot 10^{-75}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	15.7
Cost	1040

\[\begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ t_2 := \frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{if}\;t1 \leq -2.8 \cdot 10^{+36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -2.3 \cdot 10^{-25}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq -3.5 \cdot 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 1.45 \cdot 10^{-79}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	15.6
Cost	1040

\[\begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -7.4 \cdot 10^{+36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -1.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;t1 \leq -3.5 \cdot 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 1.85 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{v}{u} \cdot t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	15.3
Cost	1040

\[\begin{array}{l} t_1 := \frac{-t1}{t1 + u} \cdot \frac{v}{t1}\\ \mathbf{if}\;t1 \leq -2.5 \cdot 10^{+36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -8 \cdot 10^{-26}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \mathbf{elif}\;t1 \leq -3.5 \cdot 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 1.35 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{v}{u} \cdot t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \end{array} \]

Alternative 8
Error	15.4
Cost	1040

\[\begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ t_2 := \frac{-t1}{t1 + u} \cdot \frac{v}{t1}\\ \mathbf{if}\;t1 \leq -5.1 \cdot 10^{+38}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq -6.2 \cdot 10^{-28}:\\ \;\;\;\;t_1 \cdot \frac{t1}{u}\\ \mathbf{elif}\;t1 \leq -3.5 \cdot 10^{-145}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq 3.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{v}{u} \cdot t1}{-u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	3.2
Cost	1032

\[\begin{array}{l} t_1 := \left(-v\right) \cdot \frac{\frac{t1}{t1 + u}}{t1 + u}\\ \mathbf{if}\;t1 \leq -7 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -1 \cdot 10^{-28}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{t1 + u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	1.4
Cost	768

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

Alternative 11
Error	27.4
Cost	520

\[\begin{array}{l} t_1 := -\frac{v}{u}\\ \mathbf{if}\;u \leq -4.2 \cdot 10^{+158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 8.5 \cdot 10^{+91}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	24.8
Cost	384

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

Alternative 13
Error	30.2
Cost	256

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

Reproduce

herbie shell --seed 2023010 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))