Rosa's DopplerBench

Average Error: 17.5 → 1.3

Time: 8.0s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

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

Python

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

↓

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

Derivation

1. Initial program 17.5

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

2. Simplified 1.3

   \[\leadsto \color{blue}{\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}} \]
   Proof
   (/.f64 (/.f64 v (+.f64 t1 u)) (-.f64 -1 (/.f64 u t1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 v (+.f64 t1 u)) (-.f64 (Rewrite<= metadata-eval (neg.f64 1)) (/.f64 u t1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 v (+.f64 t1 u)) (-.f64 (neg.f64 (Rewrite<= *-inverses_binary64 (/.f64 t1 t1))) (/.f64 u t1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 v (+.f64 t1 u)) (-.f64 (Rewrite<= distribute-frac-neg_binary64 (/.f64 (neg.f64 t1) t1)) (/.f64 u t1))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 v (+.f64 t1 u)) (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (neg.f64 t1) u) t1))): 0 points increase in error, 1 points decrease in error
   (/.f64 (/.f64 v (+.f64 t1 u)) (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 t1) (neg.f64 u))) t1)): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 v (+.f64 t1 u)) (/.f64 (Rewrite=> distribute-neg-out_binary64 (neg.f64 (+.f64 t1 u))) t1)): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 v (+.f64 t1 u)) (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (+.f64 t1 u))) t1)): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 v (+.f64 t1 u)) (/.f64 (*.f64 -1 (+.f64 t1 u)) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 t1))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 v (+.f64 t1 u)) (/.f64 (*.f64 -1 (+.f64 t1 u)) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (neg.f64 t1))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 v (+.f64 t1 u)) (Rewrite=> times-frac_binary64 (*.f64 (/.f64 -1 -1) (/.f64 (+.f64 t1 u) (neg.f64 t1))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 v (+.f64 t1 u)) (*.f64 (Rewrite=> metadata-eval 1) (/.f64 (+.f64 t1 u) (neg.f64 t1)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 v (+.f64 t1 u)) (Rewrite=> *-lft-identity_binary64 (/.f64 (+.f64 t1 u) (neg.f64 t1)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 v (+.f64 t1 u)) (/.f64 (+.f64 t1 u) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 t1)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 v (+.f64 t1 u)) (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (+.f64 t1 u) t1) -1))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (/.f64 v (+.f64 t1 u)) -1) (/.f64 (+.f64 t1 u) t1))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 v (+.f64 t1 u)) (/.f64 -1 (/.f64 (+.f64 t1 u) t1)))): 11 points increase in error, 10 points decrease in error
   (*.f64 (/.f64 v (+.f64 t1 u)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 -1 t1) (+.f64 t1 u)))): 3 points increase in error, 7 points decrease in error
   (*.f64 (/.f64 v (+.f64 t1 u)) (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 t1)) (+.f64 t1 u))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= times-frac_binary64 (/.f64 (*.f64 v (neg.f64 t1)) (*.f64 (+.f64 t1 u) (+.f64 t1 u)))): 110 points increase in error, 14 points decrease in error
   (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 t1) v)) (*.f64 (+.f64 t1 u) (+.f64 t1 u))): 0 points increase in error, 0 points decrease in error

3. Final simplification 1.3

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

Alternatives

Alternative 1
Error	14.6
Cost	776

\[\begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -7.028310675683572 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 1.208154515660039 \cdot 10^{-74}:\\ \;\;\;\;\frac{t1}{\left(-u\right) \cdot \frac{u}{v}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	13.8
Cost	776

\[\begin{array}{l} t_1 := \frac{-v}{t1 + u}\\ \mathbf{if}\;t1 \leq -7.028310675683572 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 1.208154515660039 \cdot 10^{-74}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	13.6
Cost	776

\[\begin{array}{l} t_1 := \frac{-v}{t1 + u \cdot 2}\\ \mathbf{if}\;t1 \leq -7.028310675683572 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t1 \leq 1.208154515660039 \cdot 10^{-74}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{-u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	21.3
Cost	712

\[\begin{array}{l} t_1 := \frac{v \cdot \frac{t1}{u}}{u}\\ \mathbf{if}\;u \leq -3.488307489609298 \cdot 10^{+92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 5.406289180824502 \cdot 10^{+193}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	21.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;u \leq -3.488307489609298 \cdot 10^{+92}:\\ \;\;\;\;\frac{\frac{v}{u}}{\frac{u}{t1}}\\ \mathbf{elif}\;u \leq 5.406289180824502 \cdot 10^{+193}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v \cdot \frac{t1}{u}}{u}\\ \end{array} \]

Alternative 6
Error	20.7
Cost	712

\[\begin{array}{l} t_1 := t1 \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{if}\;u \leq -3.488307489609298 \cdot 10^{+92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 5.406289180824502 \cdot 10^{+193}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	20.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;u \leq -3.488307489609298 \cdot 10^{+92}:\\ \;\;\;\;t1 \cdot \frac{\frac{v}{u}}{u}\\ \mathbf{elif}\;u \leq 1.5910944293786993 \cdot 10^{+191}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \end{array} \]

Alternative 8
Error	49.9
Cost	520

\[\begin{array}{l} \mathbf{if}\;t1 \leq -3.285009030113652 \cdot 10^{+98}:\\ \;\;\;\;\frac{v}{t1}\\ \mathbf{elif}\;t1 \leq 4.8605670631510886 \cdot 10^{+75}:\\ \;\;\;\;\frac{-v}{u}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{t1}\\ \end{array} \]

Alternative 9
Error	28.0
Cost	520

\[\begin{array}{l} t_1 := \frac{-v}{u}\\ \mathbf{if}\;u \leq -3.488307489609298 \cdot 10^{+92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 1.5910944293786993 \cdot 10^{+191}:\\ \;\;\;\;\frac{-v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	25.1
Cost	384

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

Alternative 11
Error	54.6
Cost	192

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

Reproduce

herbie shell --seed 2022308 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))