Rosa's DopplerBench

Average Error: 18.2 → 1.5

Time: 7.7s

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 18.2

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

2. Simplified 1.5

   \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]
   Proof
   (*.f64 (/.f64 (neg.f64 t1) (+.f64 t1 u)) (/.f64 v (+.f64 t1 u))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (neg.f64 t1) v) (*.f64 (+.f64 t1 u) (+.f64 t1 u)))): 92 points increase in error, 9 points decrease in error

3. Final simplification 1.5

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

Alternatives

Alternative 1
Error	15.1
Cost	776

\[\begin{array}{l} t_1 := -\frac{v}{t1 + u}\\ \mathbf{if}\;t1 \leq -4 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 1.9 \cdot 10^{+34}:\\ \;\;\;\;\left(-t1\right) \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	14.2
Cost	776

\[\begin{array}{l} t_1 := -\frac{v}{t1 + u}\\ \mathbf{if}\;t1 \leq -4.4 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 7 \cdot 10^{+31}:\\ \;\;\;\;\left(-\frac{v}{u}\right) \cdot \frac{t1}{u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	21.2
Cost	712

\[\begin{array}{l} t_1 := v \cdot \frac{t1}{u \cdot u}\\ \mathbf{if}\;u \leq -8 \cdot 10^{+96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 9 \cdot 10^{+86}:\\ \;\;\;\;-\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	20.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;u \leq -7.4 \cdot 10^{+96}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{elif}\;u \leq 6.8 \cdot 10^{+86}:\\ \;\;\;\;-\frac{v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t1 \cdot \frac{v}{u \cdot u}\\ \end{array} \]

Alternative 5
Error	3.8
Cost	704

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

Alternative 6
Error	1.6
Cost	704

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

Alternative 7
Error	27.7
Cost	520

\[\begin{array}{l} t_1 := -\frac{v}{u}\\ \mathbf{if}\;u \leq -1.5 \cdot 10^{+178}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 1.45 \cdot 10^{+100}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	25.0
Cost	384

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

Alternative 9
Error	30.5
Cost	256

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

Reproduce

herbie shell --seed 2022338 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))