Rosa's DopplerBench

Average Error: 17.9 → 1.4

Time: 11.5s

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 17.9

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

2. Simplified 1.4

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

   [Start] 17.9	\[ \frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)} \]
   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	21.9
Cost	977

\[\begin{array}{l} t_1 := \frac{t1}{u} \cdot \frac{v}{u}\\ \mathbf{if}\;u \leq -5.5 \cdot 10^{+124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -8 \cdot 10^{+68}:\\ \;\;\;\;\frac{-1}{\frac{t1}{v}}\\ \mathbf{elif}\;u \leq -4.9 \cdot 10^{+43} \lor \neg \left(u \leq 6.8 \cdot 10^{+130}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]

Alternative 2
Error	21.3
Cost	976

\[\begin{array}{l} t_1 := t1 \cdot \frac{v}{u \cdot u}\\ \mathbf{if}\;u \leq -9.2 \cdot 10^{+125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq -2.4 \cdot 10^{+71}:\\ \;\;\;\;\frac{-1}{\frac{t1}{v}}\\ \mathbf{elif}\;u \leq -4.9 \cdot 10^{+43}:\\ \;\;\;\;\frac{t1}{u} \cdot \frac{v}{u}\\ \mathbf{elif}\;u \leq 2.9 \cdot 10^{+99}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	3.4
Cost	836

\[\begin{array}{l} \mathbf{if}\;u \leq 7 \cdot 10^{+189}:\\ \;\;\;\;\frac{v}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{-t1}{u}}{u}\\ \end{array} \]

Alternative 4
Error	15.0
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -1.75 \cdot 10^{-64} \lor \neg \left(t1 \leq 2 \cdot 10^{-76}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{\frac{u \cdot u}{v}}\\ \end{array} \]

Alternative 5
Error	14.0
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -2.8 \cdot 10^{-63} \lor \neg \left(t1 \leq 0.2\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{-u}\\ \end{array} \]

Alternative 6
Error	13.1
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -6.6 \cdot 10^{-64} \lor \neg \left(t1 \leq 0.028\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{u} \cdot \frac{v}{u}\\ \end{array} \]

Alternative 7
Error	21.4
Cost	713

\[\begin{array}{l} \mathbf{if}\;t1 \leq -3.1 \cdot 10^{-181} \lor \neg \left(t1 \leq 1.02 \cdot 10^{-76}\right):\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{\frac{t1}{u}}{u}\\ \end{array} \]

Alternative 8
Error	1.6
Cost	704

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

Alternative 9
Error	27.5
Cost	521

\[\begin{array}{l} \mathbf{if}\;u \leq -1.55 \cdot 10^{+159} \lor \neg \left(u \leq 4.3 \cdot 10^{+170}\right):\\ \;\;\;\;-\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]

Alternative 10
Error	30.5
Cost	256

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

Reproduce

herbie shell --seed 2023187 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))