Rosa's DopplerBench

Average Error: 17.6 → 1.4

Time: 10.7s

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 17.6

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

2. Simplified 1.4

   \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}} \]
   Proof
   (*.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)))): 1 points increase in error, 0 points decrease in error

3. Final simplification 1.4

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

Alternatives

Alternative 1
Error	13.9
Cost	1172

\[\begin{array}{l} t_1 := \frac{-v}{t1 - u}\\ t_2 := \frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{if}\;t1 \leq -5.4 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -3.45 \cdot 10^{-115}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq -2.35 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{v}{t1}}{-1 - \frac{u}{t1}}\\ \mathbf{elif}\;t1 \leq 8.4 \cdot 10^{-293}:\\ \;\;\;\;\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;t1 \leq 1.95 \cdot 10^{-73}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	13.8
Cost	1172

\[\begin{array}{l} t_1 := -1 - \frac{u}{t1}\\ t_2 := \frac{-v}{t1 - u}\\ \mathbf{if}\;t1 \leq -1.55 \cdot 10^{-26}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq -3.45 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{v}{u}}{t_1}\\ \mathbf{elif}\;t1 \leq -2.1 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{v}{t1}}{t_1}\\ \mathbf{elif}\;t1 \leq 8.5 \cdot 10^{-293}:\\ \;\;\;\;\frac{v}{\frac{u}{t1} \cdot \left(t1 - u\right)}\\ \mathbf{elif}\;t1 \leq 3.7 \cdot 10^{-73}:\\ \;\;\;\;\frac{t1 \cdot \frac{v}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	22.3
Cost	976

\[\begin{array}{l} t_1 := \frac{-v}{t1 - u}\\ t_2 := v \cdot \frac{t1}{u \cdot u}\\ \mathbf{if}\;t1 \leq -4.9 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -1.85 \cdot 10^{-84}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t1 \leq -3.6 \cdot 10^{-156}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 3 \cdot 10^{-74}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	22.3
Cost	976

\[\begin{array}{l} t_1 := \frac{-v}{t1 - u}\\ \mathbf{if}\;t1 \leq -4.9 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -2.5 \cdot 10^{-86}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{elif}\;t1 \leq -3.9 \cdot 10^{-156}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 4.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{v}{\frac{u \cdot u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	22.3
Cost	976

\[\begin{array}{l} t_1 := \frac{-v}{t1 - u}\\ \mathbf{if}\;t1 \leq -4.9 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq -5.8 \cdot 10^{-86}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{elif}\;t1 \leq -2.65 \cdot 10^{-160}:\\ \;\;\;\;\frac{v}{\left(-u\right) - t1}\\ \mathbf{elif}\;t1 \leq 4.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	13.6
Cost	777

\[\begin{array}{l} \mathbf{if}\;t1 \leq -7.5 \cdot 10^{-27} \lor \neg \left(t1 \leq 2.2 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{-v}{t1 - u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{u} \cdot \frac{-t1}{u}\\ \end{array} \]

Alternative 7
Error	1.5
Cost	704

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

Alternative 8
Error	27.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;u \leq -9.5 \cdot 10^{+168}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 9.4 \cdot 10^{+170}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1 + u}\\ \end{array} \]

Alternative 9
Error	27.6
Cost	521

\[\begin{array}{l} \mathbf{if}\;u \leq -3.8 \cdot 10^{+168} \lor \neg \left(u \leq 5 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;-\frac{v}{t1}\\ \end{array} \]

Alternative 10
Error	27.6
Cost	520

\[\begin{array}{l} \mathbf{if}\;u \leq -3.15 \cdot 10^{+168}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{elif}\;u \leq 1.35 \cdot 10^{+170}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-v}{u}\\ \end{array} \]

Alternative 11
Error	49.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;t1 \leq -1.85 \cdot 10^{+111}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 9.2 \cdot 10^{+139}:\\ \;\;\;\;\frac{v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \]

Alternative 12
Error	25.3
Cost	384

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

Alternative 13
Error	54.5
Cost	192

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

Reproduce

herbie shell --seed 2022343 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))