Rosa's DopplerBench

Average Error: 18.0 → 1.2

Time: 11.7s

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 18.0

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

2. Simplified 1.2

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

   [Start] 18.0	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   times-frac [=>] 1.2	\[ \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]

3. Final simplification 1.2

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

Alternatives

Alternative 1
Error	14.6
Cost	836

\[\begin{array}{l} \mathbf{if}\;u \leq -2.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{t1}{\frac{t1 + u}{v} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;u \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot \frac{v}{u}}{u}\\ \end{array} \]

Alternative 2
Error	14.4
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -8 \cdot 10^{-30} \lor \neg \left(t1 \leq 0.023\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{u}}{u}\\ \end{array} \]

Alternative 3
Error	13.5
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -6 \cdot 10^{-30} \lor \neg \left(t1 \leq 0.0018\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u} \cdot \frac{t1}{u}\\ \end{array} \]

Alternative 4
Error	14.8
Cost	776

\[\begin{array}{l} \mathbf{if}\;u \leq -8.2 \cdot 10^{+35}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{elif}\;u \leq 8 \cdot 10^{-11}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot \frac{v}{u}}{u}\\ \end{array} \]

Alternative 5
Error	14.3
Cost	776

\[\begin{array}{l} \mathbf{if}\;u \leq -3.2 \cdot 10^{-63}:\\ \;\;\;\;\frac{v}{t1 + u} \cdot \frac{-t1}{u}\\ \mathbf{elif}\;u \leq 3.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t1\right) \cdot \frac{v}{u}}{u}\\ \end{array} \]

Alternative 6
Error	21.5
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -6.5 \cdot 10^{+36} \lor \neg \left(u \leq 9.4 \cdot 10^{+83}\right):\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{t1}\\ \end{array} \]

Alternative 7
Error	20.6
Cost	713

\[\begin{array}{l} \mathbf{if}\;u \leq -7.1 \cdot 10^{+83} \lor \neg \left(u \leq 2.4 \cdot 10^{+85}\right):\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \end{array} \]

Alternative 8
Error	20.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;u \leq -8.2 \cdot 10^{+83}:\\ \;\;\;\;\frac{v}{\frac{u \cdot u}{t1}}\\ \mathbf{elif}\;u \leq 1.85 \cdot 10^{+86}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{u}\\ \end{array} \]

Alternative 9
Error	20.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;u \leq -5.8 \cdot 10^{+83}:\\ \;\;\;\;\frac{v}{\frac{u \cdot u}{t1}}\\ \mathbf{elif}\;u \leq 5.8 \cdot 10^{+82}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u}}{\frac{u}{t1}}\\ \end{array} \]

Alternative 10
Error	1.3
Cost	704

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

Alternative 11
Error	27.6
Cost	520

\[\begin{array}{l} \mathbf{if}\;u \leq -7.1 \cdot 10^{+83}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 3.3 \cdot 10^{+192}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u}\\ \end{array} \]

Alternative 12
Error	49.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;t1 \leq -1.4 \cdot 10^{+36}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \]

Alternative 13
Error	54.5
Cost	192

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

Reproduce

herbie shell --seed 2023027 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))