Rosa's DopplerBench

Average Error: 18.1 → 1.5

Time: 12.7s

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 18.1

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

2. Simplified 1.5

   \[\leadsto \color{blue}{\frac{\frac{v}{t1 + u}}{-1 - \frac{u}{t1}}} \]
   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))): 6 points increase in error, 0 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)))): 6 points increase in error, 9 points decrease in error
   (*.f64 (/.f64 v (+.f64 t1 u)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 -1 t1) (+.f64 t1 u)))): 7 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)))): 107 points increase in error, 15 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 1.5

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

Alternatives

Alternative 1
Error	14.9
Cost	1236

\[\begin{array}{l} t_1 := -1 - \frac{u}{t1}\\ \mathbf{if}\;u \leq -1.991215508544036 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{-t1}{u}}{\frac{t1 + u}{v}}\\ \mathbf{elif}\;u \leq 6.448278418854062 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{v}{t1}}{t_1}\\ \mathbf{elif}\;u \leq 3.5988770390465806 \cdot 10^{-12}:\\ \;\;\;\;\frac{-v}{u \cdot \frac{u}{t1}}\\ \mathbf{elif}\;u \leq 0.010240514561462605:\\ \;\;\;\;v \cdot \frac{-1}{t1}\\ \mathbf{elif}\;u \leq 4.582832012311619 \cdot 10^{+124}:\\ \;\;\;\;t1 \cdot \frac{-v}{u \cdot u}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{v}{u}}{t_1}\\ \end{array} \]

Alternative 2
Error	15.6
Cost	1104

\[\begin{array}{l} t_1 := \frac{\frac{v}{u}}{-1 - \frac{u}{t1}}\\ \mathbf{if}\;u \leq -7.164086758466106 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 6.448278418854062 \cdot 10^{-57}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq 3.5988770390465806 \cdot 10^{-12}:\\ \;\;\;\;\frac{-v}{u \cdot \frac{u}{t1}}\\ \mathbf{elif}\;u \leq 0.010240514561462605:\\ \;\;\;\;v \cdot \frac{-1}{t1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	15.6
Cost	1104

\[\begin{array}{l} t_1 := -1 - \frac{u}{t1}\\ t_2 := \frac{\frac{v}{u}}{t_1}\\ \mathbf{if}\;u \leq -7.164086758466106 \cdot 10^{+80}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;u \leq 6.448278418854062 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{v}{t1}}{t_1}\\ \mathbf{elif}\;u \leq 3.5988770390465806 \cdot 10^{-12}:\\ \;\;\;\;\frac{-v}{u \cdot \frac{u}{t1}}\\ \mathbf{elif}\;u \leq 0.010240514561462605:\\ \;\;\;\;v \cdot \frac{-1}{t1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	17.5
Cost	1040

\[\begin{array}{l} t_1 := t1 \cdot \frac{-v}{u \cdot u}\\ \mathbf{if}\;u \leq -1.991215508544036 \cdot 10^{+58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 6.448278418854062 \cdot 10^{-57}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq 3.5988770390465806 \cdot 10^{-12}:\\ \;\;\;\;\frac{-v}{u \cdot \frac{u}{t1}}\\ \mathbf{elif}\;u \leq 0.010240514561462605:\\ \;\;\;\;v \cdot \frac{-1}{t1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	15.6
Cost	1040

\[\begin{array}{l} t_1 := \frac{-t1}{u} \cdot \frac{v}{u}\\ \mathbf{if}\;u \leq -7.164086758466106 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 6.448278418854062 \cdot 10^{-57}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{elif}\;u \leq 3.5988770390465806 \cdot 10^{-12}:\\ \;\;\;\;\frac{-v}{u \cdot \frac{u}{t1}}\\ \mathbf{elif}\;u \leq 0.010240514561462605:\\ \;\;\;\;v \cdot \frac{-1}{t1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	17.4
Cost	776

\[\begin{array}{l} t_1 := t1 \cdot \frac{-v}{u \cdot u}\\ \mathbf{if}\;u \leq -1.991215508544036 \cdot 10^{+58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 0.010240514561462605:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	20.8
Cost	712

\[\begin{array}{l} t_1 := \frac{v \cdot \frac{t1}{u}}{u}\\ \mathbf{if}\;u \leq -2.365634413777459 \cdot 10^{+99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 4.685369841041647 \cdot 10^{+130}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	20.5
Cost	712

\[\begin{array}{l} t_1 := v \cdot \frac{t1}{u \cdot u}\\ \mathbf{if}\;u \leq -2.365634413777459 \cdot 10^{+99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 4.685369841041647 \cdot 10^{+130}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	20.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;u \leq -2.365634413777459 \cdot 10^{+99}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{elif}\;u \leq 6.129894139249623 \cdot 10^{+113}:\\ \;\;\;\;\frac{-v}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\frac{u \cdot u}{v}}\\ \end{array} \]

Alternative 10
Error	20.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;u \leq -2.365634413777459 \cdot 10^{+99}:\\ \;\;\;\;v \cdot \frac{t1}{u \cdot u}\\ \mathbf{elif}\;u \leq 6.129894139249623 \cdot 10^{+113}:\\ \;\;\;\;\frac{v}{u \cdot -2 - t1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t1}{\frac{u \cdot u}{v}}\\ \end{array} \]

Alternative 11
Error	27.1
Cost	584

\[\begin{array}{l} t_1 := \frac{-0.5}{\frac{u}{v}}\\ \mathbf{if}\;u \leq -1.6067313257558484 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 4.582832012311619 \cdot 10^{+124}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	27.3
Cost	520

\[\begin{array}{l} t_1 := \frac{-v}{u}\\ \mathbf{if}\;u \leq -1.6067313257558484 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u \leq 4.582832012311619 \cdot 10^{+124}:\\ \;\;\;\;-\frac{v}{t1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	24.7
Cost	384

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

Alternative 14
Error	53.8
Cost	256

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

Reproduce

herbie shell --seed 2022317 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))